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A recent paper by Villegas (1969) uses a fiducial type argument to lend support to the use of Jeffrey's invariant prior for the variance-covariance matrix in the non-informative situation. Sampling is assumed to be on a multi-normal random variable. Villegas proceeds by selecting a pivotal quantity which has a fixed distribution. Since there is nothing unique about this pivotal quantity, we note in this paper, that the Villegas fiducial approach could lead to other priors, unless more restrictions are imposed. In Section 2, we construct an example involving a "lower triangular" decomposition of a Wishart matrix variable that has the distribution of the disguised Wishart variable of Tan and Guttman (1971). Interestingly, an "upper triangular" decomposition of the same Wishart matrix leads to yet another prior.

This paper considers linear functions of order statistics of the form $S_n = n^{-1} \sum J(i/(n + 1))X_{(i)}$. The main results are that $S_n$ is asymptotically normal if the second moment of the population is finite and $J$ is bounded and continuous a.e. $F^{-1}$, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if $J$ trims the extremes. In addition, asymptotic formulas for the mean and variance of $S_n$ are given for both the identically and non-identically distributed cases. All of the theorems of this paper apply to discrete populations, continuous populations, and grouped data, and the conditions on $J$ are easily checked (and are satisfied by most robust statistics of the form $S_n$). Finally, a number of applications are given, including the trimmed mean and Gini's mean difference, and an example is presented which shows that $S_n$ may not be asymptotically normal if $J$ is discontinuous.

The purpose of this paper is to give a few theorems concerning the reciprocal relation between the convergence of a sequence of distribution functions and the convergence of the corresponding sequence of their moment generating functions. The paper consists of two parts. In the first part the univariate case is discussed. The content of this part is closely related to that of a recent paper by J. H. Curtiss [1, p. 430-433], but the results are of a somewhat more general nature, and the methods of proofs are different and do not make use of the theory of a complex variable. The second part deals with the multivariate case which, as far as the author knows, has not been treated before with proofs in as complete and rigorous a way. In both the univariate and multivariate cases the proofs are based on the well known Helly selection principle [2, p. 26] for bounded sequences of monotonic functions.

It is known that if $X$ and $Y$ are independent random variables having a Gamma distribution with parameters $(\theta, n)$ and $(\theta, m)$, i.e., with density function $p(x, \theta, n) = \frac{\theta^{n/2}x^{n/2 - 1}e^{-(\frac{1}{2})\theta x}}{2^{n/2}\Gamma(n/2)},\quad 0 < x, \theta; 1 \leqq n,$ then $X + Y$ and $X/(X + Y)$, or equivalently $X/Y$, are statistically independent. Lukacs [1] proved that this independence property characterizes the Gamma distribution, namely, if $X$ and $Y$ are two nondegenerate positive random variables, and if $X + Y$ is independent of $X/(X + Y)$, or equivalently of $X/U$, then $X$ and $Y$ have a Gamma distribution with the same scale parameter. In the present paper we present an extension to the case where $U$ and $V$ are symmetric positive definite matrices having a Wishart distribution. A number of difficulties are encountered in the generalization. First, there is no natural extension of a ratio, and we consider $Z = W^{-1}UW'$ $^{-1}$, where the "square root" $W = (U + V)^{\frac{1}{2}}$ is any factorization $WW' = (U + V)$. In the matrix case $Z$ is not a function of $V^{-\frac{1}{2}}UV^{-\frac{1}{2}}$ as was true in one dimension, and indeed if $U$ and $V$ are independent random matrices having a Wishart distribution, $U + V$ and $V^{-\frac{1}{2}}UV^{-\frac{1}{2}}$ need not be statistically independent, depending on which square root is chosen. This aspect will be treated in another paper. In the univariate case it is relatively straightforward to generate differential equations by differentiating under the expectation sign, but this is no longer true since the elements of $(U + V)^{\frac{1}{2}}$ do not bear a simple relation to the elements of $(U + V)$, and it is this point which leads to the difficulties in the proof. The characterization theorem is stated in Section 2. In Section 3 the differential equation is set up, and is solved in Section 4. The authors are indebted to Martin Fox for his comments and suggestions.

One of the most popular designs in experimental work is the randomized block. These designs can be put into three broad classes viz. complete block design, balanced incomplete block design, and the partially balanced incomplete block design. These designs are all special cases of the general two way classification with unequal numbers in the subclasses, but since the analysis of this general classification is quite complex, these special cases have evolved which are adequate to fit most needs and the analysis of these special designs is relatively easy. [1], [2], [6], [8]. However, most of the block designs considered to date have one feature in common--they require each block to contain an equal number of experimental units. The exceptions are given in [9], [10], where designs are considered in which the number of experimental units in blocks differ by one. The purpose of this paper is to extend the randomized block design to include the case where all blocks do not contain the same number of experimental units. We have called this the staircase design. Suppose an experimenter, wishing to run an experiment using $N$ treatments, decides to use a randomized block design, but after arranging his material into homogeneous groups he finds that he has blocks available which have varying number of experimental units. The experimenter has various courses open to him: (1) If enough blocks are available with $N$ or more experimental units he can discard the extra units in these blocks, discard all the blocks which have less than $N$ units, and use a randomized complete block design; (2) He can discard units in the blocks until he has enough units and blocks for a balanced incomplete block or a partially balanced incomplete block design; (3) He can use all the experimental units and use the staircase design proposed in this paper. For example, if an experimenter has $N$ treatments with which he wishes to experiment using a randomized block design, and if he has blocks of unequal size, then he must rank his $N$ treatments in the order of their importance, i.e., $T_1, T_2, \cdots, T_N,$ where he considers $T_1$ the most important and $T_N$ the least important. Now suppose he has at his disposal $b_1$ blocks which each contain $N$ experimental units. Then all $N$ treatments are randomized in each of the $b_1$ blocks. Suppose further that he has $b_2$ blocks which each contain $N_1$ experimental units $(N_1 < N).$ Then the first $N_1$ treatments are arranged at random in each of the $b_2$ blocks. This process is continued until all the blocks are used. A particular example where this would be useful is an experiment involving animals as experimental units where a block consists of litter mates. Let us suppose that we have two litters of size seven, three of size five, and one of size four. Using the staircase design we can include seven treatments and still have the four we are most interested in replicated six times.

In this paper it is shown that the optimal design for estimating any specific parameter in a polynomial spline regression with a single multiple knot at the center is supported by one of two sets of points. Familiarity with the notation and terminology used in the paper by Studden (1968) is assumed.

In this paper, the asymptotic behavior of decision procedures will be studied for a particular class of multiple decision problems. The study will throw some light on the desirability of the minimax decision procedure when the number of observations is large, and it will be seen that decision procedures frequently exist which are superior to the minimax decision procedure for large samples. The fact that the minimax decision procedure may be desirable in certain problems for small samples but undesirable for large samples was revealed by Hodges and Lehmann [1] in connection with estimation problems. Robbins [2] suggested the term "asymptotically subminimax" for the type of superior procedures which may then exist. A definition of this term which will be useful for an investigation of the asymptotic behavior of decision procedures, will be given in Section 1. A major part of this paper will be concerned with certain sequences of decision procedures called asymptotically admissible which have desirable properties similar to those of admissible decision procedures for the case of some fixed sample size. These asymptotically admissible decision procedures include a subclass of the asymptotically subminimax procedures, and the sequences of minimax procedures for those problems for which asymptotically subminimax procedures do not exist. The problems to be considered are those in which a random variable, $X$, is known to have a distribution function belonging to the distribution space, $\Omega = \{F_i(x)\}, i = 1, 2, \cdots, k$, and it is desired to select the true distribution function based on a sample of $n$ independent observations of $X$. It will be assumed that all $F_i(x)$ are absolutely continuous distribution functions having density functions, $f_i(x)$, and that for every constant $K$, the set of points for which $f_i(x)/f_j(x) = K (i \neq j)$ is a set of probability measure zero under every $F$ and all possible $i$ and $j$. A simple loss function, $W(F_i, d_j)$, where $d_j$ is the decision to select $F_j$, will be used with $W(F_i, d_j) = 1$ if an incorrect decision is made (i.e. $i \neq j)$, and $W(F_i, d_j) = 0$ if a correct decision is made (i.e. $i = j$). For such a loss function, the expected loss is simply the probability of making an incorrect decision. Section 1 will be concerned with asymptotically minimax sequences of decision procedures and Section 2 will be concerned with asymptotic admissibility. It will be seen that the asymptotic behavior of the minimax decision procedure depends on the limits of the components in the sequence of least favorable a priori distributions. Theorem 2.2 gives a sufficient condition, in terms of these limit values, for the minimax procedure to be asymptotically admissible. In Section 3, a detailed study will be made of the class of problems where $\Omega$ consists of $k$ univariate normal distributions having the same variance but different means. Let the means be denoted by $\theta_i (i = 1, 2, \cdots, k)$ with $\theta_1 < \theta_2 < \cdots < \theta_k$, and let $\min_i(\theta_{i+1} - \theta_i) = \gamma$. Then Theorems 3.1 and 3.6 will show that the minimax procedure is asymptotically admissible when the means can be put into sets, each set containing the same number, $\bar{n} \geqq 2$, of consecutive means, with a difference of $\gamma$ between any two consecutive means of a set, and a difference greater than $\gamma$ between any two means not belonging to the same set. Theorems 3.2 and 3.3 will show that in all other cases the minimax procedure is asymptotically inadmissible, and asymptotically subminimax procedures will be constructed for all these cases. Although a complete study of the asymptotic admissibility of asymptotically subminimax procedures will not be made in this paper, Theorem 3.7 will show that a certain asymptotically subminimax procedure is asymptotically admissible for all the cases covered by Theorem 3.3 and for some of the cases covered by Theorem 3.2. On the other hand, for those cases covered by Theorem 3.2 with an $\Omega$ consisting of only 3 means, it will be shown that every asymptotically subminimax procedure is asymptotically inadmissible.

The main body of this paper has two rather distinct parts, the first part (Section 2) containing the derivation of an explicit representation for the joint distribution of accident times in quite a general situation, and the remainder of the paper dealing with a very special case which leads to a discussion of testing the hypothesis that a distribution is uniform over (0, 1) against alternatives which are exponential truncated to (0, 1). The second part may be read profitably with little reference to the probabilistic arguments of Section 2.

A recent theorem of Orey [12] (see also [1], [6], [7], [13]) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.

Let $J$ be a finite non-empty set and let $S(J)$ denote the set of all finite sequences of elements of $J$. If $s = (\delta_1, \cdots, \delta_m)\in S(J)$ and $t = (\mu_1, \cdots, \mu_n)\in S(J)$, then $st$ will denote the combined sequence $(\delta_1, \cdots, \delta_m, \mu_1, \cdots, \mu_n)$. The singleton sequence $(\delta)$ will be denoted by $\delta$. The symbol $s^2$ will mean the sequence $ss$ and the symbols $s^3,s^4$, etc., are defined similarly. Suppose $\{Y_n\}$ is a stationary process with state-space $J$. If $s \in S(J)$ and has length $n, p(s)$ denotes $P\lbrack (Y_1, \cdots, Y_n) = s \rbrack$. The rank $n(\delta)$ of a $\delta \in J$ is defined to be the largest integer $n$ such that we can find $2n$ sequences $s_1, \cdots, s_n, t_1, \cdots, t_n$ in $S(J)$ such that the $n \times n$ matrix $\|p(s_i\delta t_j)\|$ is non-singular. Suppose now that $\{Y_n\}$ is a function of a finite Markov chain (hereafter abbreviated ffMc). That is, let there exist a stationary Markov chain $\{X_n\}$ with a finite state-space $I$ and a function $f$ on $I$ onto $J$ such that $\{Y_n\}$ and $\{f(X_n)\}$ have the same distribution. Then Gilbert [5] has shown that $n(\delta) \leqq N(\delta)$ for all $\delta \in J$, where $N(\delta)$ is the number of elements in $f^{-1}\lbrack\{\delta\}\rbrack$. If we can find $\{X_n\}$ and $f$ in such a way that $n(\delta) = N(\delta)$ for all $\delta \in J$, then $\{Y_n\}$ is said to be a regular ffMc. The motivation for investigating the regularity property of a ffMc has been made clear by Gilbert in the first and the last paragraphs of Section 2 of [5]. Fox and Rubin [3] have given an example of a process $\{Y_n\}$ which has $n(\delta) < \infty$ for all $\delta \in J$ but which is not a ffMc. In the first part of this paper we expand their example into a class of examples and show that some of these examples yield nonregular ffMc. These examples are of a different nature than those given in [1], Section 4. Further our method of investigation is different from that employed by Fox and Rubin. The second part of this paper is devoted to proving that an exchangeable process which is a ffMc is a regular ffMc.

Suppose we are given $N$ objects to be weighed in $N$ weighings with a chemical balance having no bias. Let $x_{ij} = 1$ if the $j$th object is placed in the left pan in the $i$th weighing, $= -1$ if the $j$th object is placed in the right pan in the $i$th weighing, $= 0$ if the $j$th object is not weighed in the $i$th weighing. The $N$th order matrix $X = (x_{ij})$ is known as the design matrix. Also let $y_i$ be the result recorded in the $i$th weighing, $\epsilon_i$ be the error in this result and $w_j$ be the true weight of the $j$th object, so that we have the $N$ equations $$x_{i1}w_1 + x_{i2}w_2 + \cdots + x_{iN}w_N = y_i + \epsilon_i, i = 1, \cdots, N.$$ We assume $X$ to be a non-singular matrix. The method of Least Squares or theory of Linear Estimation gives the estimated weights $(\hat w_i)$ by the equation $$\hat w = (X'X)^{-1}X'Y,$$ where $Y$ is the column vector of the observations and $\hat w$ is the column vector of the estimated weights. If $\sigma^2$ is the variance of each weighing, then $$\operatorname{Var} (\hat w) = (X'X)^{-1}\sigma^2 = (c_{ij})\sigma^2,$$ where $(c_{ij})$ is the inverse matrix of $(X'X)$. An expository article reviewing the work done in weighing designs is given by Banerjee [2]. Kishen [4] treats the reciprocal of the increase in variance resulting from the adoption of any design other than the most efficient design, with mean variance $\sigma^2/N$, as the efficiency of the design. This efficiency can be measured by $$1/\sum^N_{i=1} c_{ii}.$$ Mood [5] gives an alternative definition for the best weighing design. In his view the best weighing design should give the smallest confidence region in the $\hat w_i(i = 1, \cdots, N)$ space for the estimates of the weights. Hence a design will be called best if the determinant of the matrix $(c_{ij})$ is minimised. In this paper we follow Kishen's definition in obtaining the best weighing designs. Hotelling [3] proved that for the best weighing design $c_{ii} = 1/N$ and $c_{ij} = 0 (i \neq j)$. The weighing designs for which $c_{ii} = 1/N$ and $c_{ij} = 0$ are best in the sense of both Kishen and Mood. Later Mood proved that the above property is satisfied by Hadamard matrices. Plackett and Burman [6] have constructed Hadamard matrices, $H_N$, up to and including $N = 100$, excepting $N = 92$. It may be remarked here that a necessary condition for the existence of $H_N$ is $N \equiv 0 (\mod 4)$, with the exception of $N = 2$. It is not known whether this condition is sufficient or not. In this paper, the best weighing designs are obtained in the cases (i) $N$ is odd and (ii) $N \equiv 2 (\mod 4)$ subject to the conditions: i) The variances of the estimated weights are equal; and ii) The estimated weights are equally correlated. The 2nd condition here is the same as that of Banerjee [1].

In this paper, a two-sample, two-stage nonparametric estimation problem will be studied. The parameter $\theta = \theta(F, G)$ under consideration is estimable (i.e., there exists an unbiased estimator $\phi = \phi(X_1, \cdots, X_r; Y_1, \cdots, Y_s)$ of $\theta$). $\phi$ is a function of independent observations from two populations with cumulative distribution functions $F(X)$ and $G(Y)$. The functions $F(X)$ and $G(Y)$ belong to a specified class $D$, such that a $U$-statistic based on $\phi$ is the unique minimum variance unbiased estimator of $\theta$. The total number of observations on populations $X$ and $Y$ will be a fixed number $N$. The sampling procedure is carried out in two stages. First, take $M$ observations from each of the populations; then allocate the remaining $N - 2M$ observations between the populations. The method of allocation utilizes the information from the first stage observations. Two kinds of two-stage estimators, represented by $U'$, and $U'$ will be introduced in this paper. Both $U'$ and $U"$ are $U$-statistics with random sample sizes. $U'$ is based essentially on the second stage observations only. $U''$ is defined on all $N$ observations. Intuitively, the statistic $U''$ is more appealing. The first stage observations are used not only to determine the allocation of the second stage observations, but also to estimate the parameter $\theta$. (see Section 3) One of the main results (Section 4) is that $U'$ is unbiased and under certain conditions, the variance of $U'$ approaches asymptotically to a particular variance $V_0$. (Here we shall consider the cases that both the variances of $U'$ and $U''$, are finite.) $U''$ is in general biased. However, under the same conditions the value $E(U'' - \theta)^2$ approaches asymptotically to the same value $V_0$. This value $V_0$ is the smallest variance of any one-stage $U$-statistic estimator of $\theta$, subject to the restriction that the total number of observations on $X$ and on $Y$ is $N. V_0$ is computed (see Section 2) when the best one-stage allocation of $N$ observations to the two populations is made with the help of partial or even complete information about the distributions $F(X)$ and $G(Y)$. Such information about $F$ and $G$ is represented by the "nuisance parameters" $b_{10} = b_{10}(F, G), b_{01} = b_{01}(F, G)$, etc., defined in Section 2. No prior knowledge of $b_{10}$ and $b_{01}$ is required to compute $\operatorname{Var}(U')$, and $E(U'' - \theta)^2$. In Section 5, the "optimal" choice of the first stage sample size $M$ relative to the fixed total sample size $N$ is discussed. The term "optimal" is in the sense that the particular choices of $M$ in relative order of magnitude of $N$, such that as $N$ goes to infinity, the ratios $\operatorname{Var}(U')/V_0$ and $E(U'' - \theta)^2/V_0$ approach unity as fast as possible in order of magnitude of $N$. Three cases with different conditions on $\phi$ are considered. It is found that the "optimal" choices depend on the specific conditions. Section 6 contains some examples. To each $\theta(F, G)$, the corresponding estimators for $b_{10}$ and $b_{01}$ together with their behavior under different conditions on $F$ and $G$, will be given. The examples include cases where the proposed procedures can be applied as well as cases where it cannot be applied. Section 7 shows the asymptotic normality of $U'$ and $U''$. Section 8 indicates that the proposed procedures can be extended to $k$-sample case, for $k > 2$, with similar results. The technique of two-stage estimation has been used in several papers. Stein [12] used it to determine confidence interval of a pre-assigned length for the mean of a normal population with unknown variance. Putter [8] used it to estimate the mean of a stratified normal population, Robbins [10] discussed such a technic for the design of experiments. Later, Ghurye and Robbins [4] used it to estimate the difference between the means of two normal populations (or some other specified populations). Richter [9] discussed the estimation of the common mean of two normal populations. During the preparation of the present paper, Alam [1] discussed the estimation of the common mean of $k \geqq 2$ normal populations. This paper generalizes these two-stage procedures in two ways. First, the underlying cumulative distributions $F, G$ are members of a larger class of distributions. Secondly, the underlying parameters $\theta(F, G)$ are not restricted to population means or functions of means. Consequently, in such a general setup the question of "the best" estimator of any particular parameter $\theta(F, G)$ is not considered in this paper.

The following theorem is proved. Let $X_1, X_2, \cdots, X_n$ be $n$ independently (but not necessarily identically) distributed random variables, and assume that the $n$th moment of each $X_i(i = 1, 2, \cdots, n)$ exists. The necessary and sufficient conditions for the existence of two statistically independent linear forms $Y_1 = \sum^n_{s=1} a_sX_s$ and $Y_2 = \sum^n_{s=1}b_sX_s$ are: (A) Each random variable which has a nonzero coefficient in both forms is normally distributed. $(B) \sum^n_{s=1}a_sb_s\sigma^2_s = 0$. Here $\sigma^2_s$ denotes the variance of $X_s (s = 1, 2, \cdots, n)$. For $n = 2$ and $a_1 = b_1 = a_2 = 1, b_2 = -1$ this reduces to a theorem of S. Bernstein [1]. Bernstein's paper was not accessible to the authors, whose knowledge of his result was derived from a statement of S. Bernstein's theorem contained in a paper by M. Frechet [3]. A more general result, not assuming the existence of moments was obtained earlier by M. Kac [4]. A related theorem, assuming equidistribution of the $X_i (i = 1, 2, \cdots n)$ is stated without proof in a recent paper by Yu. V. Linnik [5].

In this paper a test based on the union-intersection principle is proposed for overall independence between $p$ variates distributed according to the multivariate normal law, and this is extended to the hypothesis of independence between several groups of variates which have a joint multivariate normal distribution. Methods used in earlier papers [3, 4] have been applied in order to invert these tests for each situation, and to obtain, with a joint confidence coefficient greater than or equal to a preassigned level, simultaneous confidence bounds on certain parametric functions. These parametric functions are, in case I, the moduli of the regression vectors: (a) of the variate $p$ on the variates $(p - 1), (p - 2), \cdots, 2, 1,$ or on any subset of the latter; (b) of the variate $(p - 1)$ on the variates $(p - 2), (p - 3), \cdots, 2, 1,$ or any subset of the latter, etc.; and finally, (c) of the variate 2 on the variate 1. For case II, parallel to each case considered above, there is an analogous statement in which the regression vector is replaced by a regression matrix, $\beta$, say, and the "modulus" of the regression vector is replaced by the (positive) square-root of the largest characteristic root of $(\beta\beta')$. Simultaneous confidence bounds on these sets of parameters are given. As far as the proposed tests of hypotheses of multiple independence are concerned they are offered as an alternative to another class of tests based on the likelihood-ratio criterion [5, 6] which has been known for a long time. So far as the confidence bounds are concerned it is believed, however, that no other easily obtainable confidence bounds are available in this area. One of the objects of these confidence bounds is the detection of the "culprit variates" in the case of rejection of the hypothesis of multiple independence, for the "complex" hypothesis is, in this case, the intersection of several more "elementary" hypotheses of two-by-two independence.

A vector variable $\mathbf{X}$ is said to have a linear structure if it can be written as $\mathbf{X} = \mathbf{AY}$ where $\mathbf{A}$ is a matrix and $\mathbf{Y}$ is a vector of independent random variables called structural variables. In earlier papers the conditions under which a vector random variable admits different structural representations have been studied. It is shown, among other results, that complete non-uniqueness, in some sense, of the linear structure characterizes a multivariate normal variable. In the present paper we prove a general decomposition theorem which states that any vector variable $\mathbf{X}$ with a linear structure can be expressed as the sum $(\mathbf{X}_1 + \mathbf{X}_2)$ of two independent vector variables $\mathbf{X}_1, \mathbf{X}_2$ of which $\mathbf{X}_1$ is non-normal and has a unique linear structure, and $\mathbf{X}_2$ is multivariate normal variable with a nonunique linear structure.

The present study is an extension of the previous work of the authors [3], [5], in which two randomization procedures in fractional factorial experiments were investigated. The general problem is to choose, in an optimal manner, a fractional replication of a full factorial system and an estimator, for the purpose of making inferences concerning a subset of pre-assigned parameters. We consider factorial systems of order $2^m$, which consist of $m$ factors each at 2 levels. The results can be generalized for factorial systems of order $p^m$, where $p > 2$. The factorial model adopted for the present study is the same as that adopted in the previous work, and the reader is referred to the preceeding papers [3], [5] for details and properties. The first two sections of [5] are essential for the present paper. The statistical properties of the conditional least-squares estimators (c.l.s.e.) were studied in [5] with respect to two specified randomization procedures (R.P.I. and R.P.II.), which are particular types of random allocation designs. It was shown that the c.l.s.e.'s constitute a complete class of linear unbiased estimators. The c.l.s.e. can be characterized as least-square estimators adjusted for the block of treatment combinations chosen and the information available on the nuisance parameters. In the present study we extend the investigation into the comparison of different randomization procedures. We consider a general class of procedures, characterized as follows: By some confounding method we construct $M = 2^{m-s}$ blocks, each one containing $S = 2^s$ treatment combinations. We choose one of the blocks with an arbitrary probability vector, $\xi$, and observe the associated random variables. R.P.I. is the special case where every block has the same probability of being chosen. A fixed fractional replication procedure is the special case where one of the blocks is chosen with probability one. Each randomization procedure is represented (uniquely) by a probability vector, $\xi$, and each c.l.s.e. is represented by a vector $\gamma$ in $(2^m - 2^s)$-dimensional Euclidean space. A strategy of the Statistician is thus represented by a pair of vectors $(\xi, \gamma)$. Minimax strategies are studied in the present paper for two states of information concerning the nuisance parameters: (i) all the nuisance parameters are bounded; (ii) all the nuisance parameters are bounded and all their signs are known. As proven in the present paper, the minimax strategy corresponding to Case (i) consists of R.P.I. with an unadjusted c.l.s.e. and is thus independent of the actual bounds of the nuisance parameters. On the other hand, the minimax strategy for Case (ii) consists of some fixed fractional replication with an adjusted c.l.s.e. which depends on the actual bounds of the nuisance parameters. These minimax theorems are proven with respect to a mean-square-error loss function, defined as the trace of the mean square-error matrix. A closeness loss function is considered too. The closeness of an estimator is defined as the probability that the values of the estimator will lie in a prescribed neighborhood of the true values of the parameters. In Section 2 the mean-square-error matrix and the closeness of a c.l.s.e., under an arbitrary randomization procedure, $\xi$, are derived. The mean-square-error and the closeness loss functions are defined in Section 3. Formulae for mean-square-error Bayes strategies, against any given a priori distribution of the nuisance parameters, are then derived. The closeness risk function is approximated by a similar function which has the same Bayes strategies as the mean-square-error loss function. It is shown that when the size of the experiment, $S = 2^s$, grows the closeness risk function and the approximative function converge. The minimax theorems are stated and proved in Section 4 only with respect to the mean-square-error loss function. It can be shown that the minimax closeness strategies, for the states of information studied, are the same as those for the mean-square-error loss function. This can be concluded also from the results of Section 3. In Section 5 we present a numerical example to illustrate the results and the computations involved.

Let $x_{i1}, x_{i2}, \cdots, x_{ini}$ be a random sample of real observations from the $i$th population with cumulative distribution function (cdf) $F_i(x), i = 1,2, \cdots, c$. Let the $c$ samples be independent and the $F$'s continuous. In this paper we shall consider tests for the null hypothesis $H_0:F_1(x) = F_2(x) = \cdots = F_c(x) = F(x), \text{say}$. The statistics and tests, proposed in this paper, are based upon $c$-plets of observations which are formed by selecting one observation from each of the $c$ samples. The total number of distinct $c$-plets that can be formed in this way is $\prod^c_{i=1}n_i$. In each $c$-plet we compare and rank observations appearing therein. Let $v_{ij}$ be the number of $c$-plets in which the observation selected from the $i$th sample is larger than exactly $(j - 1)$ observations and smaller than the other $(c - j)$ observations. Since the distributions are assumed to be continuous the probability of the existence of ties is zero. Let us define $u_{ij} = \nu_{ij}/\prod^c_{i=1}n_i$; it is the proportion of $c$-plets which give rank $j$ to the observation from the $i$th sample. Let us have $N = \sum^c_{i=1}n_i, p_i = n_i/N, L_i = \sum^c_{j=1} a_ju_{ij}$, where the $a$'s are real constants such that they are not all equal and \begin{equation*} \tag{1.1} A = \sum^c_{j=1} \sum^c_{l=1} a_ja_l \big\{\frac{\binom{c-1}{j-1}\binom{c-1}{l-1}}{(2c - 1)\binom{2c-2}{j+l-2}} - \frac{1}{c^2}\big\}.\end{equation*} Then we define a class of statistics $\mathscr{L}$ as \begin{equation*} \tag{1.2} \mathscr{L} = \frac{N(c - 1)^2}{Ac^2} \big\lbrack \sum^c_{i=1} p_iL_i^2 - \big(\sum^c_{i=1} p_i L_i\big)^2 \big\rbrack.\end{equation*} A particular member of the class is found by specifying the real constants $a$'s. With each member of this class we associate a test of $H_0$: Reject $H_0$ at a significance level $\alpha$ if $\mathscr{L}$ exceeds some predetermined constant $\mathscr{L}_\alpha$. We, later in this paper, show that under $H_0, \mathscr{L}$ is distributed as a $\chi^2$ variate with $c - 1$ degrees of freedom, in the limit as $N \rightarrow \infty$. Hence for sufficiently large $N, \mathscr{L}_\alpha$ may be approximated by the corresponding significance point of the $\chi^2$ distribution with requisite degrees of freedom. Tests proposed by Bhapkar [2], [3], Sugiura [13], and the author [5], [6] may be seen to belong to this class. In this paper it is attempted to provide a unified treatment of statistics and tests based on $c$-plets--particularly those based on linear combinations of the $u$'s. The detailed properties of statistics belonging to this class are discussed under the null hypothesis and the following two alternative hypotheses. (I) the alternative of different locations or shift, the distributions being equal in all other respects and, (II) the alternative of different scales, the distributions again being equal in all other respects. Haller [7] has discussed the use and the properties of some statistics belonging to this class for testing $H_0$ against an alternative of stochastically ordered variables and for selection and ranking procedures. In the fourth section we give a condition on the distributions under which these tests are consistent against specified alternatives. In the fifth section $\mathscr{L}$ is shown to have a limiting noncentral $\chi^2$ distribution with $c - 1$ degrees of freedom under the pertinent alternative hypotheses. The noncentrality parameter is seen to be a quadratic form in the constants $a$'s, involving $F$. The earlier test statistics, mentioned above, were constructed taking into account the relative magnitudes of the $u$'s under the null and under the alternative hypotheses. The idea was to emphasize the difference between the two magnitudes. This "difference" is, in some sense, maximized if we are able to obtain the statistics, from the class, which has the largest noncentrality parameter under the alternative hypothesis of interest. This statistic would then be recommended to test $H_0$ whenever the particular alternative is suspected as likely. Also, for this particular alternative hypothesis, this test shall have maximum asymptotic relative efficiency (in the Pitman sense) among the class of statistics proposed. In the sixth section we show how to obtain the statistics with the above property and do so for certain specified alternatives. In the same section we compute the ARE of these tests with respect to certain of their competitors.

Let $(\mathscr{X}, \mathscr{a})$ be a measurable space and $\Theta$ be an open subset of a $k$-dimensional Euclidean space. For each $\theta \varepsilon \Theta$ let $P_\theta$ be a probability measure on $\mathscr{a}$. We assume that for every $\theta \varepsilon \Theta, \{X_n, n \geqq 0\}$ is a Markov process defined on $(\mathscr{X}, \mathscr{a})$ into $(R, \mathscr{B})$, where $(R, \mathscr{B})$ denotes the Borel real line. Furthermore, we denote by $\mathscr{a}_n$ the $\sigma$-fields induced by the random variables $\{X_0, X_1, \cdots, X_n\}$, and by $P_{n, \theta}$ the restriction of $P_\theta$ to $\mathscr{a}_n$. In the present paper we give conditions under which the sequence of families of probability measures $\{P_{n,\theta}, \theta \varepsilon \Theta\}$ has the desirable property of being differentially asymptotically normal. This implies that in some neighborhood of $\theta \varepsilon \Theta, \{P_{n,\theta}, \theta \varepsilon \Theta\}, n \geqq 0$ can be, for certain problems, approximately treated as if they were normal. For a detailed account of the notions involved in this paper the reader is referred to [5], in particular, Section 5. Also in the Appendix of the present paper one can find the definitions of the concepts most frequently used in this work, including that of the differentially asymptotically normal families of distributions. In Section 2 the required notation is introduced and also the assumptions being made throughout the paper are listed. In Section 3 we state the main result, and in the following subsections we give the proof of it in several steps.

0. Summary. This paper studies the product epsilon entropy of mean-continous Gaussian processes. That is, a given mean-continuous Gaussian process on the unit interval is expanded into its Karhulnen expansion. Along the kth eigenfunction axis, a partition by intervals of length Ek iS made, and the entropy of the resulting discrete distribution is noted. The infimum of the sum over k of these entropies subject to the constraint that E Ek _ = is the product epsilon entropy of the process. It is shown that the best partition to take along each eigenfunction axis is the one in which 0 is the midpoint of an interval in the partition. Furthermore, the product epsilon entropy is finite if and only if E Xk log Xk-' is finite, where Xk is the kth eigenvalue of the process. When the above series is finite, the values of Ek which achieve the product entropy are found. Asymptotic expressions for the product epsilon entropy are derived in some special cases. The problem arises in the theory of data compression, which studies the efficient representation of random data with prescribed accuracy. 1. Introduction. This paper is motivated by the problem of data compression, the efficient representation of data for the purpose of information transmission. We shall consider the case in which the data to be represented consists of a sample function from a Gaussian process X (t) on the unit interval which is mean-continuous; i.e. E[x (s) - x (t)]2 > 0 as s -> t, for all t. Our basic problem is how to transmit (over a noiseless channel) information as to which sample function of X occurred. We assume that the recipient of the transmitted data has full knowledge of the statistics of the process. In particular he knows the Karhiinen expansion [1] of the process; namely

This paper studies the problem of testing the independence of two random variables $X, Y$ from a random sample, $(X_1, Y_1), (X_2, Y_2) \cdots (X_n, Y_n)$, of size $n$, where $X$ and $Y$ are angular variates (i.e., reals modulo 1). In the standard case where $X$ and $Y$ are ordinary real variables, the following approach has been useful. Suppose $(X, Y)$ has the continuous distribution function $F(x, y)$ with marginal distribution functions $F_1(x)$ and $F_2(y)$, respectively. It is desired to test $H_0: F(x, y) = F_1(x)F_2(y)$ against the alternative $H_A: F(x, y) \neq F_1(x)F_2(y)$. Let $F_n(x, y)$ denote the sample distribution function of the random bivariate sample, i.e., if $H(x)$ denotes the left continuous Heaviside function then \begin{equation*}\tag{1}F_n(x, y) = \frac{1}{n} \sum^n_{j=1} H(x - X_j)H(y - Y_j).\end{equation*} Also, let $F_{n1}(x)$ and $F_{n2}(y)$ denote the sample distribution functions associated with the first and second components of the random sample vector. In terms of $H(x)$ \begin{equation*}\tag{2}F_{n1}(x) = \frac{1}{n} \sum^n_{j=1} H(x - X_j)\end{equation*} and \begin{equation*}\tag{3}F_{n2}(y) = \frac{1}{n} \sum^n_{j=1} H(y - Y_j).\end{equation*} Blum, Kiefer and Rosenblatt [1] studied the following distribution free tests of independence based on the sample distribution function. Reject for large values of \begin{equation*}\tag{4}A_n = \sup_{x,y}|T_n(x, y)|\end{equation*} or \begin{equation*}\tag{5}B_n = n \int\int\lbrack T_n(x, y)\rbrack^2 dF_n(x, y),\end{equation*} where \begin{equation*}\tag{6}T_n(x, y) = F_n(x, y) - F_{n1}(x)F_{n2}(y).\end{equation*} The first statistic, constructed in the spirit of Kolomorov-Smirnov statistics, has good power properties, cf. [1], Section 4, but its asymptotic distribution is unknown. The statistic $B_n$ is analogous to the Cramer-von Mises statistic and is also equivalent to a statistic originally proposed by Hoeffding [3]. The characteristic function of the null asymptotic distribution of $B_n$ is (cf. [1] and [3]) \begin{equation*}\tag{7}E e^{izB} = \prod^\infty_{j,k=1} \big(1 - \frac{2iz}{\pi^4j^2k^2}\big)^{-\frac{1}{2}}.\end{equation*} Asymptotic power properties are given in [1]. As in Rothman [9] the difficulty in modifying these tests in the toroidal case is that there is no natural origin for the distribution functions on a circle. Moreover, different arbitrary starting points give the test statistics $A_n$ and $B_n$ different values. In this paper we propose that the statistic $C_n$ be used for our problem, with the surface of the torus replacing the plane. \begin{equation*}\tag{8}C_n = n \int\int Z_n^2(x, y) dF_n(x, y)\end{equation*} where $Z_n(x, y) = T_n(x, y) + \int\int T_n(x, y) dF_n(x, y) - \int T_n(x, y) dF_{n1}(x) - \int T_n(x, y) dF_{n2}(y).$ We note that $C_n$ may be rewritten in the following form which we shall refer to in Section 3: \begin{equation*}\tag{9}C_n = 1/n^2 \sum^n_{j=1} \lbrack\sum^n_{a=1} \{T_n(X_j, Y_j) + T_n(X_a, Y_a) - T_n(X_a, Y_j) - T_n(X_j, Y_a)\}\rbrack^2.\end{equation*} We shall also have occasion to use the random variable, $D_n = n \int\int Z_n^{\ast^2}(x, y) dF_1(x)dF_2(y),$ where $Z_n^\ast(x, y) = T_n(x, y) + \int\int T_n(x, y)dF_1(x)dF_2(y) - \int T_n(x, y)dF_1(x) - \int T_n(x, y) dF_2(y).$ An outline of the paper follows: In Section 2 it is shown that when $H_0$ is true, $C_n - D_n \rightarrow 0$ in probability. The invariance of $C_n$ under changes of origin is proved in Section 3. Finally the asymptotic distribution of $C_n$ under the null hypothesis is obtained in Section 4.

We define the incomplete gamma function ratio for positive $N$ by \begin{align*}\tag{1}P(N, b, X) &= \int^X_0 t^{N-1}e^{-bt} dt/ \int^\infty_0 t^{N-1}e^{-bt} dt \\ &\equiv \int^X_0 D(N, b, t) dt, \text{say},\\ \end{align*} where $N$ and $b$ are positive real numbers and $0 < X \leqq \infty$. In what follows, we generally use the notation $P_N$ to denote the same function, unless $b$ or $X$ are given some specific values. It should be noted that the notation $P(N, b, X)$ does not mean that the function is one of three variables, since we have $P(N, b, X) = P(N, 1, bX)$. The positive real number $b$ is only a scale factor. The function $P_N$ is of appreciable importance in probability and statistics and it is known as the gamma distribution and for $b = \frac{1}{2}$ as the chi-square distribution. It is also of importance in many other branches of applied mathematics. Like the incomplete beta distribution, it is also a special case of the more general confluent hypergeometric function. During the last decade, interest has been revived in the development, use and application of these functions and a number of publications, such as Erdelyi et al. (1953), Tricomi (1954) and Slater (1960), appeared. In these publications, the function treated is defined as the numerator in the middle part of (1) with $b = 1$, and the definition is generally extended for all complex values of $N$ such that $R(N)$ is not a non-positive integer. Many of the properties of $P_N$ are derived in these publications. Bancroft (1949) derived some new properties of the incomplete beta function (particularly, recurrence relations) from the more general properties of the parent hypergeometric function. Similar methods may also be used to derive new properties of $P_N$ of use in statistical work. In this paper, however, we derive such properties directly from the well-known difference-differential properties of $P_N$ given by \begin{align*}\tag{2}(\partial^r/\partial X^r) P_N &= (-b)^r\Delta^rP_{N-r}, \\ &= (\partial^{r-1}/\partial X^{r-1})D(N, b, X),\quad N > r,\\ \end{align*} where $r = 0, 1, 2, 3, \cdots$ and where \begin{equation*}\tag{3}\Delta^{r+1} P_T = \Delta^r(\Delta P_T) = \Delta^r(P_{T+1} - P_T)\end{equation*} are the advancing finite differences of $P_N$ with respect to $N$ and with a unit difference interval. Property (2) for $r = 1$ may be used, in fact, to define the function $P_N$ itself, as in Milne-Thomson (1933). This property was also used in 1947 by Khamis in an unpublished Ph.D thesis (1950) to derive, inter alia, a series expansion in terms of chi-square integrals to approximate to statistical distribution functions and also to formulate a computational method for the tabulation of the chi-square distribution itself. This method was later used by Hartley and Pearson (1950) in computing a five decimal table for this distribution. The chi-square series expansion was later generalized by Khamis (1960a) into an expansion in terms of incomplete gamma function ratios and the computational method was employed by Khamis, (1964a) and (1965) to prepare a six decimal table of the chi-square integral and a master ten decimal table, for very fine $N$ and $X$ intervals, of the incomplete gamma function (1) for $b = \frac{1}{2}$ (a description of this table is given by Khamis (1964b). In this paper we make further use of the property (2) to derive other new and useful properties of the function (1). In Section 2 we derive a simple recurrence relation for $P(T, b, X)$ for $T = N, N + 1$ and $N + 2$. In Section 3 we derive new $N$-wise sum formulae as consequences of this recurrence relation. We then give in Section 4 an $X$-wise sum formula based on the Taylor expansion of (2) in the neighbourhood of $X$. This is made possible by extending the definition of $P(N, b, X)$ for $N \leqq 0$, and thus enabling the removal of the condition $N > r$ in (2) above. Other examples of the use of property (2) are given in Section 5 where in particular, the Laguerre polynomials are expressed in terms of the differences of the function $P(N, b, X)$. Section 5 contains also examples of some numerical applications of the previous results. The importance of the properties derived here and the simplicity inherent in such derivations due to the nature of property (2) are further enhanced by work recently carried out by Wise (1950) and developed further by H. O. Hartley and E. J. Hughes (in process of publication) where the incomplete gamma function ratio is shown to provide quite satisfactory approximations to the incomplete beta function ratio and to many other statistical functions. We note finally, that the notation used in (1) is only one of many notations used by different authors. The references given above include most of the different notations used for the function $P_N$ and other related functions.

Let $Z_1, Z_2, \cdots, Z_N$ be $N$ random variables with common distribution $P(Z_i \leqq u) = F(u - \theta)$ where $F \varepsilon \mathscr{F}_{cs0}$ which throughout this paper means the class of all continuous distributions symmetric about zero. The distributions considered will furthermore satisfy the regularity conditions of Lemma 3a in [3]. $\theta$ is an unknown constant to be estimated. In the case where $Z_1, \cdots, Z_N$ are independent, the following estimate of $\theta$ has been recently investigated by J. L. Hodges Jr. and E. L. Lehmann [4]. \begin{equation*}\tag{0.1}\theta^{\ast} = \operatorname{med}_{r\leqq s} \{(Z_r + Z_s)/2\},\end{equation*} .e. the median of the $N + \binom{N}{2}$ averages $(Z_r + Z_s)/2$. The asymptotic efficiency of $\theta^{\ast}$ relative to the classical estimate \begin{equation*}\tag{0.2}\hat{\theta} = \sum^N_{i=1} Z_i/N\end{equation*} in the sense of reciprocal ratio of asymptotic variances has been determined in [4] and shown to be $12\sigma^2_z\lbrack\int f{}^2(z) dz\rbrack^2$ where $f$ is the density corresponding to $F$ and $\sigma_z^2$ denotes the variance of the $Z$'s. It follows directly from Theorem 2.2 of [6] that $\theta^{\ast}$, in case of independent but not necessarily symmetrically distributed observations $Z_j$, is a consistent estimate of the pseudomedian of $F$ ([6], p. 178), which in general may be different from the median $\theta$. In this paper we shall consider a situation where only few $(c)$ observations can be collected per day and where the experiments have to be conducted over several $(n)$ days to yield the necessary number of observations. During this period the experimental conditions may easily change, whereby the standard assumption of "independent and identically distributed" observations is violated. The data occur naturally grouped in $n$ blocks, $c$ observations per block, and the possible change of conditions is introduced as a (nuisance) random block effect. We shall study the behavior of the two estimates $\theta^{\ast}$ and $\hat\theta$ under such conditions to find out how robust they are against this kind of dependence. In particular we shall study their asymptotic behavior as $n \rightarrow \infty$ with $c$ fixed, and shall derive a general expression for the asymptotic efficiency of $\theta\ast$ relative to $\hat{\theta}$. The efficiency is finally computed for normal and gross error models.

Two models are considered in detail which are the Models I and II of ANOVA in the terminology of Eisenhart [2]. The present paper, which deals with the one dimensional or univariate case, and its sequel, which will deal with the multidimensional or multivariate case, seek to give a unified general treatment, using matrix methods, of certain problems under the two models of ANOVA. Section 1 of each paper, which deals with Model I, is of the nature of a resume giving the main results of a general treatment discussed elsewhere [10, 11, 12] by one of the authors. Section 2 of each paper, which deals with Model II or variance components model, is self-contained, and presents a natural tie-up between the analyses under the two models for a $k$-way classification. Results in estimation, testing of hypotheses and confidence bounds are presented, although the main emphasis is on the results in confidence bounds (simultaneous and/or separate) on meaningful parametric functions which are physically natural and mathematically convenient measures of departure from customary null hypotheses. It will be seen that a mixed model, which would include both Models I and II as special cases, can be defined, and the associated problems can be studied by using methods which are, essentially, a combination of the methods given for the separate models in Sections 1 and 2, respectively, of this paper. Since nothing essentially new is involved in such a study, this paper does not explicitly discuss it. Unless otherwise stated, capital letters will denote matrices and small letters in boldface will denote column vectors. Such letters when primed denote transposes. For instance, $A(p \times q)$ denotes a matrix with $p$ rows and $q$ columns, $A'(q \times p)$ denotes the transpose of $A, \mathbf{a}(p \times 1)$ denotes a column vector with $p$ elements and $\mathbf{a}'(1 \times p)$, the transpose of $\mathbf{a}$, is a row vector. In particular, $I(p)$ will denote the identity matrix of order $p$ and $0(p \times 1)$ and $0(p \times q)$ will stand, respectively, for the null vector of order $p$ and the null matrix with $p$ rows and $q$ columns.

Let $\{Z(t); t \geqq 0\}$ be a one dimensional age dependent branching process with offspring probability generating function (pgf) $h(s) \equiv \sum^\infty_{j=0} p_js^j$ and lifetime distribution function $G(t)$ (see Section 2 for definitions). If $m(t) \equiv EZ(t)$ is the mean function let $Y(t) = Z(t)/m(t)$. Our objective in this paper is to study the limiting behavior of the process $\{Y(t); t \geqq 0\}$. The main result is THEOREM 0. Assume $Z(0) \equiv 1, m = h'(1) > 1, G(0+) = 0$. (Here $\rightarrow_p$ and $\rightarrow_d$ mean convergence in probability and distribution respectively). Then: \begin{equation*}\tag{1}\sum^\infty_{j=2} j \log jp_j = \infty\quad\text{implies} Z(t)/EZ(t) \rightarrow_p 0\end{equation*} and \begin{equation*}\tag{2}\sum^\infty_{j=2} j \log jp_j < \infty\quad\text{implies} Z(t)/EZ(t) \rightarrow_d W\end{equation*} where $W$ is an nonnegative random variable such that (a) $EW = 1$, (b) $\varphi(u) = E(e^{-uW})$ for $u \geqq 0$ satisfies \begin{equation*}\tag{3}\varphi(u) = \int^\infty_0 h(\varphi(ue^{-\alpha y})) dG(y)\end{equation*} where $\alpha$ is the unique root of the equation $m \int^\infty_0 e^{-\alpha y} dG(y) = 1$ (c) $P(W = 0) = q$ the extinction probability (d) $W$ has an absolutely continuous distribution on the positive real axis and the density function is continuous. That is, there exists a nonnegative continuous function $g(x)$ defined for $x > 0$ such that for $0 < x_1 < x_2 < \infty$ \begin{equation*}\tag{4}P(x_1 < W < x_2) = \int^{x_2}_{x_1} g(x) dx.\end{equation*} Kesten and Stigum [4] proved the above result for the case when $G(x)$ is the step function \begin{equation*}\tag{5}\begin{align*}G(x) = 0 \text{if} x\leqq 1 \\ = 1\quad x > 1.\end{align*}\end{equation*} This is the Galton-Watson process in discrete time. They considered the multi-dimensional case. Athreya and Karlin [1] considered the case (here $0 < \lambda < \infty)$ \begin{equation*}\tag{6}\begin{align*}G(x) = 1 - e^{-\lambda x} \text{for} x > 0 \\ = 0\quad x \leqq 0.\end{align*}\end{equation*} This is the continuous time Markov branching process. Their approach was via split times. Levinson [6] established the law convergence of $Z(t)/EZ(t)$ under conditions slightly stronger than ours. Harris [3] claimed mean square convergence of $Z(t)/EZ(t)$ when $h'' (1) < \infty$ and the absolute continuity of $W$ when in addition to $h'' (1) < \infty, 1 - G(t) = O(e^{-cr})$ for some $c > 0$. Our result is the sharpest known in this direction in as much as (i) we establish the convergence of $Z(t)/EZ(t)$ without any conditions, (ii) we give a necessary and sufficient condition for the nondegeneracy of the limit random variable $W$ and (iii) when $W$ is nondegenerate we establish the absolute continuity without any extra assumptions. The methods employed in this paper are all extremely simple. Among them are a simplified and sharpened form of Levinson's [6] arguments and a simplification of Stigum's [7] idea to prove absolute continuity of $W$. One of the important ideas used here is the exploitation of the underlying Galton-Watson process constituted by the size $\{\zeta_n\}$ of the different generations. The key to the understanding of the moment condition $\sum_j j \log jp_j < \infty$ is the simple Lemma 1. Here is an outline of the rest of the paper. In Section 2 we describe the setting and introduce the necessary terminology and notation. The functional equation (3) is studied in detail in Section 3 where it is shown that a necessary and sufficient condition for (3) to have a nontrivial solution is the finiteness of $\sum j \log jp_j$. The next section explores the connection between the process $\{Z(t); t \geqq 0\}$ and the underlying Galton-Watson process $\{\zeta_n; n = 0, 1, 2, \cdots\}$ and shows that if $\sum j \log jp_j = \infty$ then $Z(t)/EZ(t) \rightarrow_p 0$. Assuming $\sum j \log jp_j < \infty$ the convergence in distribution of $Z(t)/EZ(t)$ to a nondegenerate random variable $W$ is shown in Section 5 while Section 6 takes up the proof of absolute continuity. The last section lists some open problems.

Let $X_{ij} (j = 1, 2,\cdots, n_i; i = 1, 2,\cdots, k)$ be $k$ independent samples of sizes $n_1, n_2,\cdots, n_k$ respectively from $k$ populations with continuous cumulative distribution functions $F_1, F_2,\cdots, F_k$ respectively. We assume that the $F_i$'s belong to a family $\mathscr{F}$ of distribution functions indexed by a parameter $\theta$. Let all the $N = \sum^k_{i=1} n_i$ observations be put together and ordered to form a single sequence and suppose that only the first $r$ ordered observations are available. That is, let us have a combined (right) censored sample of total size $r$. Such a censored sample occurs naturally in many physical situations as for example, in problems of life testing where we are interested in comparing the mean life of several physical systems, or in clinical trials or bio-assay problems where we want to compare the efficacy of several drugs but we cannot afford to wait indefinitely to get information on all the sampling units put on test. For details see Basu [2]. For facility of discussion, we shall use the terminology of life testing. Any test based on the first $r$ ordered observations (out of a combined sample of size $N$) will be termed an $r$ out of $N$ test. In this paper we propose two $k$-sample $r$ out of $N$ rank tests which generalize the rank tests proposed by Kruskal [7], Jonckhere [6] and Terpstra [10], [11]. In the first part of the paper we propose the statistic $B_r^{(N)}$ (large values being critical) to test the null hypothesis \begin{equation*}\tag{1.1}H_0:F_1(x) = F_2(x) = \cdots = F_k(x)\end{equation*} (or equivalently, $H_0:\theta_1 = \theta_2 = \cdots = \theta_k = 0$ say, under location alternatives) against the alternative hypothesis \begin{equation*}\tag{1.2}H_1:F_i(x) = F(x, \theta_i) \quad (i = 1, 2, \cdots, k)\end{equation*} In Section 2 we define the statistic $B_r^{(N)}$ and show its relationship with other statistics. The mean and variance of $B_r^{(N)}$ under the null hypothesis is derived in Section 3. In Section 4 we find the asymptotic distribution of $B_r^{(N)}$ both under the null and the non-null case. The computation of $B_r^{(N)}$ has been illustrated by an example in Section 5. In the second part of the paper we consider the statistic $V(N, r)$ (to be defined later) for testing the hypothesis (1.1) against the ordered alternative \begin{equation*}\tag{1.3}H_2:F_1(x) < F_2(x) < \cdots < F_k(x)\end{equation*} for all $x$, where the $F_i(x)$ are labeled in such a way that (1.3) is the ordered alternative being considered. The second formulation is useful in life testing where the $k$ cdf's might be associated with $k$ different processes and the experimenter wishes to test whether the $k$ processes give rise to units with the same life-time distributions against the alternative that the processes can be ordered in a particular manner, in the sense that the $k$ life-time distributions can be ordered uniformly with respect to $x$, (that is, according to their reliabilities). In Section 6 we define the statistic $V(N, r)$ and in Section 7 we have derived the mean and variance of $V(N, r)$ under $H_0$. Section 8 is devoted to investigate the extreme values of $V(N, r)$ and the asymptotic normality of $V(N, r)$ is proved in Section 9. Both the statistics $B_r^{(N)}$ and $V)N, r)$ may be considered as $k$-sample extensions of the $V_r$ statistic proposed by Sobel [9]. The definitions of $B_r^{(N)}$ and $V(N, r)$ are slightly different from what is given earlier in Basu [2].

This paper concerns the following problem posed by R. Pyke (1958). What is the cardinality, $M_k$, of the maximal family of stochastically independent random variables defined on a given space $\Omega$, of cardinality ${\bar{\bar{\Omega}}} = k$? Since maximality is sought, the investigation is limited to two-valued, nontrivial (tvnt) random variables; and the $\sigma$-algebra of measurable subsets of $\Omega$ is taken to be that generated by the family of random variables. With these restrictions the problem is essentially one of cardinality. The results are summarized in the table below. Theorem 1 follows from the fact that stochastic independence entails the non-vanishing of certain finite intersections of elementary sets. Theorem 2 is a result of Kakutani, Kodaira and Oxtoby [4, 5, 6]. Theorem 3 is a consequence of a set theoretic result of Tarski [11], and a theorem of Banach [1, 2, 10, 11], which results were used in proofs of Theorem 2. Theorem 4 follows from a construction and a lemma of Marczewski [9]. The paper is divided into five sections. Section 1 introduces the notation and terminology. Section 2 discusses two types of independence. Sections 3, 4, and 5 treat, respectively, the finite, non-countable and countable cases.

The purpose of this paper is to compare for various two-parameter distributions, of the form $f\{(x - \mu)/\sigma\}/\sigma,$ the estimates of the parameters obtained by applying the method of least squares to the observations, after these have been arranged in order of magnitude. Estimates obtained by this process we shall call "ordered least-squares estimates." Such estimates are unbiased and have minimal variance among all unbiased estimates which are linear in the ordered observations. This estimation process has been previously discussed by Godwin [1] and [2] and Lloyd [3]. In the present paper, ordered estimates are obtained explicitly for a class of two-parameter distributions having the above form. This class contains the rectangular and the right triangular distributions as special cases. It also reduces to the exponential distribution as a limiting case. Other special cases of this class of distributions have also been previously discussed by Craig [4]. Further, a general property of ordered least-squares estimates of the parameter $\lambda$ in distributions of the type $f(x/\lambda)/\lambda$ is discussed. As a result it is shown that the ordered least-squares estimate of the scale parameter in the Pearson Type III distribution is identical with the maximum likelihood estimate.

Associated with certain of the learning models introduced by Bush and Mosteller [1] are random walks $p_1, p_2, p_3, \cdots$ on the closed unit interval with transition probabilities of the form \begin{equation*}\tag{1}P\lbrack p_{n + 1} = p_n + \theta_1(1 - p_n) \mid p_n\rbrack = \varphi(p_n)\end{equation*} and \begin{equation*}\tag{2}P\lbrack p_{n + 1} = p_n - \theta_2p_n \mid p_n\rbrack = 1 - \varphi(p_n)\end{equation*} where $0 < \theta_1, \theta_2 < 1$ and $\varphi$ is a mapping of the closed unit interval into itself. In the experiments to which these models are applied, response alternatives $A_1$ and $A_2$ are available to a subject on each of a sequence of trials, and $p_n$ is the probability that the subject will make response $A_1$ on trial $n$. Depending on which response is actually made, one of two events $E_1$ or $E_2$ ensues. These events are associated, respectively, with the increment $p_n \rightarrow p_n + \theta_1(1 - p_n)$ and the decrement $p_n \rightarrow p_n - \theta_2p_n$ in $A_1$ response probability. The conditional probabilities $\pi_{ij}$ of event $E_j$ given response $A_i$ do not depend on the trial number $n$. Thus (1) and (2) are obtained with $\varphi(p) = \pi_{11}p + \pi_{21}(1 - p)$. Since the linearity of the functions $\varphi$ which arise in this way is of no consequence for the work presented in this paper, we will assume instead simply that \begin{equation*}\tag{3}\varphi \varepsilon C^2(\lbrack 0, 1\rbrack).\end{equation*} We impose one further restriction on $\varphi$ which excludes some cases of interest in learning theory: \begin{equation*}\tag{4}\epsilon_1 = \min_{0 \leqq p \leqq 1} \varphi(p) > 0 \text{and} \epsilon_2 = \max_{0 \leqq p \leqq 1} \varphi(p) < 1.\end{equation*} It follows from a theorem of Karlin ([5], Theorem 37) that under (1)-(4) the distribution function $F^{(n)}_{\theta_1,\theta_2,\varphi}$ of $p_n$ (which depends, of course, on the distribution $F$ of $p_1$) converges as $n$ approaches infinity to a distribution $F_{\theta_1,\theta_2,\varphi}$ which does not depend on $F$. It is with the distributions $F_{\theta_1,\theta_2,\varphi}$ that the present paper is concerned. Very little is known about distributions of this family, though some results may be found in Karlin [5], Bush and Mosteller [1], Kemeny and Snell [6], Estes and Suppes [3], and McGregor and Hui [8]. The only theorem in the literature directly relevant to the present work is one of McGregor and Zidek [9] as a consequence of which, in the case $\theta_1 = \theta_2 = \theta, \varphi(p) \equiv \frac{1}{2}$, $\lim_{\theta \rightarrow 0} \lim_{n \rightarrow \infty} P\lbrack\theta^{-\frac{1}{2}}(p_n - \frac{1}{2}) \leqq x\rbrack = \Phi(8^{\frac{1}{2}}x)$ where $\Phi$ denotes the standard normal distribution function; that is, the distribution $F_{\theta,\theta,\frac{1}{2}}(\theta^{\frac{1}{2}}x + \frac{1}{2})$ converges to a normal distribution as the "learning rate" parameter $\theta$ tends to 0. We will prove, by means of another method, that this phenomenon is of much greater generality. Theorem 1 below shows that, for any positive constant $\zeta$ and any $\varphi$ with $\max_{0 \leqq p \leqq 1}\varphi'(p) < \min (1, \zeta)/\max (1, \zeta)$ there is a constant $\rho$ such that $F_{\theta,\zeta\theta,\varphi}(\theta^{\frac{1}{2}}x + \rho)$ converges to a normal distribution as $\theta \rightarrow 0$. A nonnormal limit is obtained if $\theta_1$ approaches 0 while $\theta_2$ remains fixed as is shown in Theorem 2. In this case $F_{\theta_1,\theta_2,\varphi}(\theta_1x)$ converges to an infinite convolution of geometric distributions. If $f(p,\theta) = p + \theta(1 - p)$ then (1) and (2) can be written in the form \begin{equation*}\tag{5}P\lbrack p_{n + 1} = f(p_n, \theta_1) \mid p_n\rbrack = \varphi(p_n)\end{equation*} and \begin{equation*}\tag{6}P\lbrack p_{n + 1} = 1 - f(1 - p_n, \theta_2) \mid p_n\rbrack = 1 - \varphi(p_n).\end{equation*} In Section 4 it is shown that the linearity of $f(p,\theta)$ in $p$ and $\theta$ is not essential to the phenomena discussed above. Theorems 3 and 4 present generalizations of Theorems 2 and 1, respectively, to "learning functions" $f(p,\theta)$ subject only to certain fairly weak axioms. A somewhat different learning model, Estes [2] $N$-element pattern model, leads to a finite Markov chain $p_1, p_2, p_3, \cdots$ with state space $S_N = \{jN^{-1}: j = 0, 1, \cdots, N\}$ and transition probabilities \begin{align*}\tag{7}P\lbrack p_{n + 1} &= p_n + N^{-1} \mid p_n\rbrack = \varphi(p_n), \\ \tag{8}P\lbrack p_{n + 1} &= p_n - N^{-1} \mid p_n\rbrack = \psi(p_n),\\ \end{align*} and \begin{equation*}\tag{9}P\lbrack p_{n + 1} = p_n \mid p_n\rbrack = 1 - \varphi(p_n) - \psi(p_n)\end{equation*} where $\varphi(p) = c\pi_{21}(1 - p), \psi(p) = c\pi_{12}p, 0 < c \leqq 1$, and for the sake of this discussion we suppose that $0 < \pi_{12}, \pi_{21} < 1$. In this case a limiting distribution $F_{N, \varphi,\psi}$ of $p_n$ as $n \rightarrow \infty$ exists and is independent of the distribution of $p_1$ by a standard theorem on Markov chains. Estes [2] showed that the limit is binomial over $S_N$ with mean $r = \pi_{21}/(\pi_{12} + \pi_{21})$. It then follows from the central limit theorem that $\lim_{N \rightarrow \infty}\lim_{n \rightarrow \infty} P\lbrack N^{\frac{1}{2}}(p_n - r) \leqq x\rbrack = \Phi\lbrack x/(r(1 - r))^{\frac{1}{2}}\rbrack.$ In Section 5 it is shown that our method permits an extension of this result to much more general $\varphi$ and $\psi$.

In this paper we have studied certain combinatorial properties of incomplete block designs and efficient necessary conditions for the existence of affine resolvable balanced incomplete block (b.i.b., for abbreviation) designs. Two theorems give combinatorial properties of certain b.i.b. designs. The well known inequality of Fisher between the number of varieties and number of blocks is shown in this paper to hold under very general conditions. An intrinsic characteristic property of symmetrical b.i.b. designs is given in another theorem. In the last two theorems we have deduced efficient necessary conditions for the existence of affine resolvable b.i.b. designs. Besides these there are some minor results. Utilizing the simple yet very fruitful idea of associating an incidence matrix with a design, all the results are deduced with the help of arguments of algebra of matrices and linear equations. The last theorem requires the use of the celebrated four square theorem of Lagrange and a result due to Legendre in number theory.

Motivated by ecological and genetic phenomena, Karlin and McGregor [3] introduced the following model to describe the continued formation and growth of mutant biological populations. It is assumed that a new mutant population arises at each event time of a stochastic process (referred to as the input process) $\{v(t),t > 0\}$ whose state space is the non-negative integers. Each new mutant population begins its evolution with a fixed number of elements and evolves according to the laws of a continuous time Markov chain $\mathscr{P}$ with stationary transition probability function $P_{i,j}(t)\quad i,j = 0, 1, 2, \cdots; t \geqq 0.$ We assume that all populations evolve according to the same Markov Chain and independent of one another. In terms of this structure, the basic question which we consider in this work can be formulated in the following manner: (A) Given an input process $\{v(t),t > 0\}$ and the individual growing process $\mathscr{P}$, determine the asymptotic behavior as $t \rightarrow \infty$ of the mean and variance of $S(t) = \{$number of different sizes of mutant populations at time $t\}$ and determine the limit distribution as $t \rightarrow \infty$ of $S(t)$ appropriately normalized. $S(t)$ is a special functional of the vector process $N(t) = \{N_0(t), N_1(t), N_2(t), \cdots\}\quad t > 0$ where $N_k(t) = \text{number of mutant populations with exactly} k \text{members at time} t$ and may be interpreted as a measure of the fluctuations in population size. We have restricted our considerations to this special case because it serves as a model problem for more general situations and possesses all the subtle difficulties of the general case. The random variable $S(t)$ can also be identified as the number of distinct occupied states at time $t$ among all Markov Chains which have begun their evolution up to that time. In the subsequent discussion we will refer to $\{S(t), t > 0\}$ as the "occupied states" process generated by the input process $\{v(t),t > 0\}$ and the Markov Chain $\mathscr{P}$. Without loss of generality we identify the state 0 as the initial state of all evolving Markov Chains and $-1$ as an absorbing state if absorption is possible. In this paper we introduce "occupied states" processes generated by a class of null recurrent, transient, and absorbing barrier Birth and Death processes and a Poisson input process. The special feature of this class is that with the normalization $Y(t;u) = t^{-1} X(t^\alpha u),\quad\alpha > 0$ $(\{X(t), t > 0\}$ is the growing process $\mathscr{P}$), the process $Y(t;u)$ converges weakly in the Markov sense as $t \rightarrow \infty$ to a Bessel diffusion (see C. Stone [8]). The main idea (also applicable to more general growing processes $\mathscr{P}$ is that one requires local limit theorems, and under some circumstances, specification of the rate of convergence of the transition density of $Y(t;u)$ to the density of the limiting diffusion in order to prescribe exact asymptotic formulas for $ES(t)$ and $\operatorname{Var} S(t)$ and to prove a central limit theorem for $S(t)$. The results of this paper are in sharp contrast with the asymptotic formulas for $ES(t)$ and $\operatorname{Var} S(t)$ which appear in the companion paper [6], where the growing process $\mathscr{P}$ is a general positive recurrent Markov chain and the input process remains Poisson. Section 2 contains basic definitions, some intuitive discussion, and precise statements of the main results on asymptotic behavior of $ES(t)$ and $\operatorname{Var} S(t)$. In Section 3, we present detailed proofs of the theorems of Section 2, and we conclude with a central limit theorem for $S(t)$ in Section 4. The appendix contains some technical lemmas which are essential for asymptotic formulas that incorporate speed of convergence theorems.

This paper deals with the problem of estimating the number of trials of a multinomial distribution, from an incomplete observation of the cell totals, under constraints on the cell probabilities. More specifically let $(n_1, \cdots, n_k)$ be distributed according to the multinomial law $M(N; p_1, \cdots, p_k)$ where $N$ is the number of trials and the $p_i$'s are the cell probabilities, $\sum^k_{i=1}p_i$ being equal to 1. Suppose that only a proper subset of $(n_1, \cdots, n_k)$ is observable, that $N, p_1, \cdots, p_k$ are unknown and that $N$ is to be estimated. Without loss of generality, $(n_1, \cdots, n_{l-1}), l \leqq k$ may be taken to be the observable random vector. For fixed $N, (n_1, \cdots, n_{l-1}, N - n)$ has the multinomial distribution $M(N; p_1, \cdots, p_l)$ where $n$ denotes $\sum^{l-1}_{i=1}n_i$ and $p_l$ denotes $1 - \sum^{l-1}_{i=1}p_i$. If the parameter space is such that $N$ can take any nonnegative integral value and each $p_i$ can take any value between 0 and 1, such that $\sum^{l-1}_{i=1}p_i < 1$ then, clearly, the only inference one can make about $N$ is that $N > n$. In specific situations, it might, however, be possible to postulate constraints of the type \begin{equation*}\tag{1.1} p_i = f_i(\theta),\quad i = 1, \cdots, l\end{equation*} where $\theta = (\theta_1, \cdots, \theta_r)$ is a vector of $r$ independent parameters and $f_i$ are known functions. This may lead to estimability of $N$. The problem of estimating $N$ in such a situation is studied here. The present investigation is motivated by the following problem. Experiments in particle physics often involve visual scanning of film containing photographs of particles (occurring, for instance, inside a bubble chamber). The scanning is done with a view to counting the number $N$ of particles of a predetermined type (these particles will be referred to as events). But owing to poor visibility caused by such characteristics as low momentum, the distribution and configuration of nearby track patterns, etc., some events are likely to be missed during the scanning process. The question, then, is: How does one get an estimate of $N$? The usual procedure of estimating $N$ is as follows. Film containing the $N$ (unknown) events is scanned separately by $w$ scanners (ordered in some specific way) using the same instructions. For each event $E$ let a $w$-vector $Z(E)$ be defined, such that the $j$th component $Z_j$ of $Z(E)$ is 1 if $E$ is detected by the $j$th scanner and is 0 otherwise. Let $\mathscr{J}$ be the set of $2^w w$-vectors of 1's and 0's and let $I_0$ by the vector of 0's. Let $x_I$ be the number of events $E$ whose $Z(E) = I$. For $I \in \mathscr{J} - \{I_0\}$, the $x_I$'s are observed. A probability model is assumed for the results of the scanning process. That is, it is assumed that there is a probability $p_I$ that $Z(E)$ assumes the value $I$ and that these $p_I$'s are constrained by equations of the type (1.1) (These constraints vary according to the assumptions made about the scanners and events, thus giving rise to different models. An example of $p_I(\theta)$ would be $E(\nu^{\Sigma^w_{j=1}I_j}(1 - \nu)^{w-\Sigma^w_{j=1}I_j})$ where $I_j$ is the $j$th component of $I$ and expectation is taken with respect to the two-parameter beta density for $v$. This is the result of assuming that all scanners are equally efficient in detecting events, that the probability $v$ that an event is seen by any scanner is a random variable and that the results of the different scans are locally independent. For a discussion of various models, see Sanathanan (1969), Chapter III. $N$ is then estimated using the observed $x_I$'s and the constraints on the $P_I$'s, provided certain conditions (e.g., the minimum number of scans required) are met. The following formulation of the problem of estimating $N$, however, leads to some systematic study including a development of the relevant asymptotic distribution theory for the estimators. The $Z(E)$'s may be regarded as realizations of $N$ independent identically distributed random variables whose common distribution is discrete with probabilities $p_I$ at $I$ (In particle counting problems, it is usually true that the particles of interest are sparsely distributed throughout the film on account of their Poisson distribution with low intensity. Thus in spite of the factors affecting their visibility outlined earlier, the events can be assumed to be independent.). The joint distribution of the $x_I$'s is, then, multinomial $M(N; p_I, I \in \mathscr{J})$. The problem of estimating $N$ is now in the form stated at the beginning of this section. Since the estimate depends on the constraints provided for the $p_I$'s, it is important to test the "fit" on the model selected. The conditional distribution of the $x_I$'s $(I \neq I_0)$ given $x$ is multinomial $M(x; p_I/p(I \neq I_0))$ where $x$ is defined as $\sum_{I\neq I_0} x_I$ and $p$ as $\sum_{I\neq I_0}P_I$. The corresponding $\chi^2$ goodness of fit test may therefore be used to test the adequacy of a model in question. Various estimators of $N$ are considered in this paper and among them is, of course, the maximum likelihood estimator of $N$. Asymptotic theory for maximum likelihood estimation of the parameters of a multinomial distribution has been developed before for the case where $N$ is known but not for the case where $N$ is unknown. Asymptotic theory related to the latter case is developed is Section 4. The result on the asymptotic joint distribution of the relevant maximum likelihood estimators is stated in Theorem 2. A second method of estimation considered is that of maximizing the likelihood based on the conditional probability of observing $(n_1,\cdots, n_{l-1})$, given $n$. This method is called the conditional maximum likelihood (C.M.L.) method. The C.M.L. estimator of $N$ is shown (Theorem 2) to be asymptotically equivalent to the maximum likelihood estimator. Section 5 contains an extension of these results to the situation involving several multinomial distributions. This situation arises in the particle scanning context when the detected events are classified into groups based on some factor like momentum which is related to visibility of an event, and a separate scanning record is available for each group. A third method of estimation considered is that of equating certain linear combinations of the cell totals (presumably chosen on the basis of some criterion) to their respective expected values. Asymptotic theory for this method is given in Section 6. This discussion is motivated by a particular case which is applicable to some models in the particle scanning problem, using a criterion based on the method of moments for the unobservable random variable, given by the number of scanners detecting an event (Discussion of the particular case can be found in Sanathanan (1969) Chapter III.). In the next section we give some definitions and a preliminary lemma.

A factorial experiment involving $m$ factors such that the $i$th factor has $m_i$ levels is termed an asymmetrical factorial design. If the number of levels is equal to one another the experiment is termed a symmetric factorial experiment. When the block size of the experiment permits only a sub-set of the factorial combinations to be assigned to the experimental units within a block, resort is made to the theory of confounding. With respect to symmetric factorial designs, the theory of confounding has been highly developed by Bose [1], Bose and Kishen [4], and Fisher [11], [12]. An excellent summary of the results of this research appears in Kempthorne [13]. However, these researches are closely related to Galois field theory resulting in (i) only symmetric factorial designs being incorporated into the current theory of confounding; (ii) the common level must be a prime (or power of a prime) number; and (iii) the block size must be a multiple of this prime number. The theory of confounding for asymmetric designs has not been developed to any great degree. Examples of asymmetric designs can be found in Yates [19], Cochran and Cox [9], Li [15], and Kempthorne [13]. Nair and Rao [16] have given the statistical analysis of a class of asymmetrical two-factor designs in considerable detail. Kramer and Bradley [14] discuss the application of group divisible designs to asymmetrical factorial experiments, however their paper is mainly confined to the two-factor case and its intra-block analysis. It is the purpose of this paper, which was done independently of their work, to outline the general theory for using the group divisible incomplete block designs for asymmetrical factorial experiments. The use of incomplete block designs for asymmetric factorial experiments results in (i) no restriction that the levels must be a prime (or power of a prime) number, (ii) no restriction with respect to the dependence of the block size on the type of level, and (iii) unlike the previous referenced works on asymmetric factorial designs, the resulting analysis is simple, does not increase in difficulty with an increasing number of factors, and "automatically adjusts" for the effects of partial confounding. Section 2 states three useful lemmas, Section 3 contains the main results of this paper, and Section 4 outlines the recovery of inter-block information.

In an earlier paper [4], the author has given upper bounds for the number of disjoint blocks in (i) Semi-regular GD designs, (ii) certain PBIB designs with two associate classes having triangular association scheme, (iii) certain PBIB designs with two associate classes having $L_2$ association scheme and (iv) certain PBIB designs with three associate classes having rectangular association scheme. In this paper, we give bounds for the number of common treatments between any two blocks of the above-mentioned PBIB designs. The main tools used to establish the results of this paper are the theorems proved by (i) Bose and Connor [1], (ii) Raghavarao [3], and (iii) Vartak [6].

The multiplicative process is usually defined by the sequence of random variables $X_0, X_1, \cdots$ whose distributions are specified as follows: $P(X_0 = 1) = 1, \sum^\infty_{\nu=0} P(X_1 = \nu) = 1,$ and if $X_n = 0$ then $P(X_{n+1} = 0) = 1,$ whereas if $X_n$ is a positive integer then $X_{n+1}$ is distributed as the sum of $X_n$ independent random variables each with the distribution of $X_1$. The variable $X_n$ is interpreted as the number of "particles" in the $n$th generation, and the index $n$ as a discrete time parameter. This has been the method of approach in previous studies of the process [1, 2, 3, 4, 5]. The multiplicative process has various applications, notably in the study of population growth, the spread of epidemics of rumors, and the nuclear chain reaction. The closely related "birth and death" process was recently studied by Kendall [6]. Whenever one studies the probability theory of a particular system there seem to be definite conceptual advantages in defining explicitly the set $\mathcal{J}$ of elementary events, the additive class $\mathcal{M}$ of subsets of $\mathcal{J}$, called events, and the probability measure $P$ for the events of $\mathcal{M}$. Now an elementary event of this process can be represented by a rooted tree where the original particle is represented by the root vertex and where the particles of the $n$th generation are represented by the vertices $n$ segments removed from the root. The tree will be finite or infinite according to whether a finite or an infinite number of particles are involved in the elementary event. Thus, the set of trees is the natural choice for $\mathcal{J}$. The first part of this paper is devoted to a more precise description of $\mathcal{J, M}$ and $P$. We shall then see easily that $X_n(t)$, the number of vertices $n$ segments removed from the root of $t \epsilon \mathcal{J}$, i.e. the number of particles in the $n$th generation, has the distribution defined in the preceding paragraph. Since the time does not appear in our description of $\mathcal{J}$ we fetter ourselves somewhat if we interpret $n$ as a discrete time parameter. Thus, we have already reaped some harvest from considering the process from the point of view of $\mathcal{J}$. Another advantage is that we are led in a natural way to study the distribution of other structural features of the trees, e.g. the total number of vertices, or the number of vertices with $k$ outgoing segments. The chief results of this paper are as follows. The recursion formula for the probability $P_n$ that a tree have $n$ vertices $n = 1, 2, \cdots$ is obtained as well as an asymptotic estimate of $P_n$ valid for large $n$. The distributions of the number of branches at the root in a finite tree, an infinite tree, or in a tree with $n$ vertices are obtained and the asymptotic distribution of the latter as $n \rightarrow \infty$. The distribution of the fraction of vertices with $k$ outgoing segments in the finite trees, in the trees with $n$ vertices, and the asymptotic distribution of the latter as $n \rightarrow \infty$ are also found. Finally, an estimate is obtained for the probability that a tree be finite in case this probability is near 1, a result which was previously obtained by Kolmogoroff [7].

This paper deals with linear regressions \begin{equation*}\tag{1.1}y_k = x_{k1}\beta_1 + \cdots + x_{kq}\beta_q + \epsilon_k, \quad k = 1, 2, \cdots\end{equation*} with given constants $x_{km}$ and with error random variables $\epsilon_k$ that are (a) uncorrelated or (b) independent. Let $E\epsilon_k = 0, 0 < E\epsilon^2_k < \infty$ for all $k$. The individual error distribution functions (d.f.'s) are not assumed to be known, nor need they be identical for all $k$. They are assumed, however, to be elements of a certain set $F$ of d.f.'s. Consider the family of regressions associated with the family of all the error sequences possible under these restrictions. Then conditions on the set $F$ and on the $x_{km}$ are obtained such that the least squares estimators (LSE) of the parameters $\beta_1, \cdots, \beta_q$ are consistent in Case (a) (Theorem 1) or asymptotically normal in Case (b) (Theorem 2) for every regression of the respective families. The motivation for these theorems lies in the fact that under the given assumptions statements based only on the available knowledge must always concern the regression family as a whole. It will be noticed moreover that the conditions of the theorems do not require any knowledge about the particular error sequence occurring in (1.1). Most of the conditions are necessary as well as sufficient, with the consequence that they cannot be improved upon under the limited information assumed to be available about the model. Since the conditions are very mild, the results apply to a large number of actual estimation problems. We denote by $\mathfrak{F}(F)$ the set of all sequences $\{\epsilon_k\}$ that occur in the regressions of a family as characterized above. Thus, $\mathfrak{F}(F)$ comprises all sequences of uncorrelated (Case (a)) or independent (Case (b)) random variables whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. For each $G \varepsilon F$ the relations $\int x dG = 0$ and $0 < \int x^2 dG < \infty$ hold. In this paper, $\mathfrak{F}(F)$ may be looked upon as a parameter space. A parameter point then is a sequence of $\mathfrak{F}(F)$. Correspondingly, we say that a statement holds on $\mathfrak{F}(F)$ (briefly on $F$) if it holds for all $\{\epsilon_k\} \varepsilon \mathfrak{F}(F)$. The statements of Theorems 1 and 2 are of this kind. The proof of Theorem 1, as well as the proof of the sufficiency in Theorem 2, is elementary and straight forward. Theorem 2 is a special case of a central limit theorem (holding uniformly on $\mathfrak{F}(F)$) for families of random sequences [3]. Some similarity between the roles of the parameter spaces $\mathfrak{F}(F)$ in our theorems and of the parameter spaces that occur, e.g., in the Gauss-Markov and related theorems may be seen in the fact that these theorems remain true only as long as the conclusions in the theorems hold for every parameter point in the respective spaces. As is well known, the statements in the Gauss-Markov and related theorems hold for every parameter vector $\beta_1, \cdots, \beta_q$ in a $q$-dimensional vector space (see e.g. Scheffe 1959, p. 13, 14). A result in the theory of linear regressions that bears some resemblance with the theorems of this paper has been obtained by Grenander and Rosenblatt (1957, p. 244). Let the error sequence $\{\epsilon_k\}$ in (1.1) be a weakly stationary random sequence with piecewise continuous spectral density, and let the regression vectors admit a joint spectral representation. Under these assumptions Grenander and Rosenblatt give necessary and sufficient conditions for the regression spectrum and for the family of admissible spectral densities in order that the LSE are asymptotically efficient for every density of the family. In Sections 3 and 6 we discuss some examples relevant to Theorems 1 and 2.

In this paper the asymptotic expansion of a percentage point of Hotelling's generalized $T^2_0$ distribution is derived in terms of the corresponding percentage point of a $\chi^2$ distribution. Our result generalizes Hotelling's and Frankel's asymptotic expansion for the generalized Student $T$ [3], [4]. The technique used in this paper for obtaining the asymptotic expansion of $T^2_0$ is an extension of the previous methods of Welch [8] and of James [5], [6], who used them to solve the distribution problem of various statistics in connection with the Behrens-Fisher problem. An asymptotic formula for the cumulative distribution function (c.d.f.) of $T^2_0$ is also given together with an upper bound for the error committed when all but the first few terms are omitted in the series. This formula is a sort of multivariate analogue of Hartley's formula of "Studentization" [2].

This paper continues a study, initiated in [1], of the stochastic independence of $V = X - Y$ and the mean of order $\gamma$, \begin{equation*}\tag{1}M_\gamma\begin{align*} = (1/\gamma) \log \lbrack(e^{\gamma X} + e^{\gamma Y})/2\rbrack\quad\text{for} \gamma \neq 0 \text{and} \gamma \neq \pm \infty\\ = (X + Y)/2\quad\text{for} \gamma = 0\\ = \max (X, Y)\quad\text{for} \gamma = + \infty\\ = \min (X, Y)\quad \text{for} \gamma = - \infty\end{align*}\end{equation*} for independent $X$ and $Y$. The case $\gamma = 0$ is solved by the well known result, if, for independent $X$ and $Y$, the variables $X + Y$ and $X - Y$ are independent, then both $X$ and $Y$ have normal distributions with a common variance. The first proof of this result may be found in Kac [6]. (Although the statement of this result there is much weaker, Kac's proof applies exactly to the statement above.) A corresponding result for the gamma distribution, proved in restricted form by Hogg [5], and proved without restrictions by Lukacs [7], is as follows: if $Z_1$ and $Z_2$ are independent positive random variables for which $Z_1 + Z_2$ and $Z_1/Z_2$ are independent, then both $Z_1$ and $Z_2$ have gamma distributions with a common scale parameter. This result may be used to complete the study of the independence of $M_\gamma$ and $V$ for independent $X$ and $Y$ when $\gamma$ is finite and non-zero, by letting $Z_1 = \exp (\gamma X)$ and $Z_2 = \exp (\gamma Y)$. It is clear, then, from the result of Hogg and Lukacs that both $\exp (\gamma Y)$ and $\exp (\gamma Y)$ must have gamma distributions with a common scale parameter. The resulting distributions of $X$ and $Y$ have been studied in [1]. Therefore, of this study, there remain yet to be completely solved only the cases $\gamma = +\infty$ and $\gamma = - \infty$. These two cases are essentially the same since if $M_{+\infty}$ and $V$ are independent, then so are $\min (-X, -Y)$ and $V$. Thus, the negatives of the distributions for which $M_{+\infty}$ and $V$ are independent will yield independent $M_{-\infty}$ and $V$. We consider only the case $\gamma = -\infty$. This problem is considered in a previous paper, [2], in which the distributions were restricted to be discrete. The main result of that paper is as follows: if $X$ and $Y$ are independent, non-degenerate, discrete random variables, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then $X$ and $Y$ both have geometric distributions with common location and scale parameters, but possibly different geometric parameters. If the probability mass function of the geometric distribution is written as \begin{equation*}\tag{2}p(x) = (1 - p)p^{(x-\theta)/c}\quad\text{for} x = \theta, \theta + c, \theta + 2c, \cdots\end{equation*} then, the parameter $\theta$ is a location parameter, $c$ is a scale parameter, and $p$ is called the geometric parameter, $c > 0, 0 < p < 1$. In this paper, the distributions of $X$ and $Y$ are restricted to be absolutely continuous. This will result in a characterization of the so-called exponential distribution whose density is \begin{equation*}\tag{3}f(x) = (1/\sigma) \exp (-(x - \theta)/\sigma)\quad\text{if} x > \theta\end{equation*} and zero otherwise. The parameter $\theta$ is a location parameter, and the parameter $\sigma > 0$ is a scale parameter, which is also the standard deviation of the distribution. In these terms the main result of this paper may be stated as follows: if $X$ and $Y$ are independent random variables with absolutely continuous distributions, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then both $X$ and $Y$ have exponential distributions with a common location parameter but with possibly different scale parameters. This result then gives a characterization of the exponential distribution, since it is easy to show the converse, that if $X$ and $Y$ are independent and if each has an exponential distribution with a common location parameter but with possibly different scale parameters, then $U$ and $V$ are independent. For related results, the reader is referred to papers by Fisz [3] and Rogers [8].

1. Preliminaries. Let ,u be a given probability measure on (X, B), where X is some finite dimensional Euclidean space and B is the class of Borel sets on X. For each 0 E X, let Il6(A) = p(A-0), where A C B and A-0 = {x: x+0 c A}. A set 'A' is called ,-invariant if ilo(A) = p(A) V 0 E X. For example the null set 0 and the whole space X are trivially p-invariant. The class of all p-invariant sets is denoted by A(,u). A set 'A' is called "non-trivial" if 0 < ,l(A) < 1. That "nontrivial" p-invariant sets exist is seen by noting that if ,I assigns probability I each to {O} and {1}, then A is ,u-invariant if and only if A' = A+1, e.g., A = -O (2n, 2n+ 1] is p-invariant with p(A) = 2. It is easily seen that A(,) is a monotone class, is closed for complementation and disjoint unions, and is not necessarily a r-algebra. The probability measure ,u is called weakly incomplete (weaklycomplete) if it has (or does not have) non-trivial It-invariant sets. The results we present here have originated from a paper of Basu and Ghosh (1969) on p-invariant sets. Our main object is to make a careful study of some of the conjectures contained in their paper. A brief account of the results contained in the paper is as follows

This study was initiated in connection with estimating parameters involved in a certain stochastic process of population growth. Because of the nature of distribution functions arising in such studies, the usual methods of estimation result in formulas which are so complex that it is difficult, if not impossible, to obtain explicit solutions for the estimates of the parameters. Investigation of the problem led to an extension of the method of best asymptotically normal estimates developed by Neyman [1]. The estimates derived are termed regular best asymptotically normal estimates (RBAN estimates). This extension can be applied to other problems. In [1], Neyman considers a whole class of estimates which possess the properties of consistency, of asymptotic normality, and of asymptotic efficiency, and he provides estimates having these asymptotic properties for the case of multinomial distributions. His method is extended in the present paper to a more general case in which random vectors are dealt with. Such an extension was considered by Barankin and Gurland [2], who studied a large class of estimates and showed that if the distributions involved are members of Koopman's family, it is still possible to reach the Cramer-Rao lower bound. The purposes of the present paper are to discuss a subclass of the estimates considered by Barankin and Gurland and to present simple methods of generating such estimates. The estimates discussed are based on a number of independent random vectors whose distribution functions are not specified. It is proved that under certain regularity conditions, the regular and consistent estimates obtained are asymptotically normal as the number of random vectors tends to infinity. A necessary and sufficient condition for a regular and consistent estimate to have a "minimal" asymptotic covariance matrix is given. An expression is derived for the "minimal" asymptotic covariance matrix. It is also proved that if a function $\mathbf{f}$ satisfies certain conditions, then in order that $\mathbf{f}(\tilde\theta)$ be an RBAN estimate of $\mathbf{f}(\theta)$ at $\mathbf{f}(\theta^0)$, where $\theta^0$ is the true value of the parameter point $\theta$, it is necessary and sufficient that the argument $\tilde\theta$ be an RBAN estimate of $\theta$ at $\theta^0$. Methods of generating RBAN estimates are given. For simplicity of presentation, matrix notation is used throughout this paper. By derivatives of a matrix with respect to a vector (or with respect to a second matrix) is meant the derivatives of the matrix simultaneously with respect to all the components of the vector (or all the elements of the second matrix). The usual rules of differentiation with respect to vectors are used.

Let $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e. $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P(|x_{nk}| > \epsilon) = 0$ for every $\epsilon > 0$, and for each $n, x_{n1}, \cdots, x_{nk_n}$ are independent). Let $S_n = x_{n1} + \cdots + x_{nk_n} - A_n$ where the $A_n$ are constants and let $F_n(x)$ be the distribution function of $S_n$. In a previous paper [3] the system of infinitesimal, rowwise independent random variables $(|x_{nk}|^r)$ was studied for $r \geqq 1$. Specifically, let $S^r_n = |x_{n1}|^r + \cdots + |x_{nk_n}|^r - B_n(r),$ where the $B_n(r)$ are suitably chosen constants. Let $F_n^r(x)$ be the distribution function of $S^r_n$. Necessary and sufficient conditions for $F_n^r(x)$ to converge $(n \rightarrow \infty)$ to a distribution function $F^r(x)$ and for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ were given, together with the form that $H(x)$ must take. In Section 2 of this paper we consider the system $(|x_{nk}|^r)$ for $0 < r < 1$. Results similar to the above are found, replacing $(r \rightarrow \infty)$ by $(r \rightarrow 0^+)$. However different assumptions must be made at certain points. Various remarks are made in this paper to show where the results here differ from [3]. In particular it is shown that, if $F^r(x)$ converges $(r \rightarrow 0^+)$ to a distribution function $H(x)$, then $H(x)$ will be the distribution function of the sum of two independent random variables, one Poisson and the other Gaussian. Furthermore, while the Gaussian summand may or may not be degenerate, the Poisson summand will be nondegenerate in all but one special case.

The present investigation is concerned with the formulation of a multivariate approach for the construction of a class of aligned rank order tests for the analysis of variance (ANOVA) problem relating to two-way layouts. The problems of simultaneous testing and testing for ordered alternatives based on aligned rank order statistics are also considered. Various efficiency results pertaining to the proposed tests are studied. Let us consider a two factor experiment comprising $n$ blocks, each block containing $p(\geqq 2)$ plots receiving $p$ different treatments. In accordance with the two-way ANOVA model, we express the yield $X_{ij}$ of the plot receiving the $j$th treatment in the $i$th block as \begin{equation*}\tag{1.1}X_{ij} = \mu + \alpha_i + \tau_j + \epsilon_{ij}, \quad i = 1, \cdots, n, j = 1, \cdots p;\end{equation*} where $\mu$ stands for the mean effect, $\alpha_1, \cdots, \alpha_n$ for the block effects (may or may not be stochastic), $\tau_1, \cdots, \tau_p$ for the treatment effects (assumed to be non-stochastic), and $\epsilon_{ij}$'s are the residual error components. It is assumed that $\mathscr{\epsilon}_i = (\epsilon_{i1}, \cdots, \epsilon_{ip}), i = 1, \cdots, n$ are independent and identically distributed stochastic vectors having a continuous (joint) cumulative distribution function (cdf) $G(x_1, \cdots, x_p)$ which is symmetric in its $p$ arguments; (this includes the conventional situation of independence and identity of distributions of all the $np$ error components as a special case). We may set without any loss of generality $\sum^p_1 \tau_j = 0$, and frame the null hypothesis of no treatment effect as \begin{equation*}\tag{1.2}H_0 : \tau_1 = \cdots = \tau_p = 0.\end{equation*} The usual ANOVA test based on the variance-ratio criterion is valid only when $G$ is a $p$-variate (totally symmetric) multinormal cdf. For arbitrarily continuous cdf $G(x_1, \cdots, x_p)$, intra-block rank tests are due to Friedman [7], Brown and Mood [3], and Sen [21]; generalizations of these tests to incomplete layouts are due to Durbin [6], Benard and Elteren [1], and Bhapkar [2]. Hodges and Lehmann [9] have pointed out that intra-block rank tests do not utilize the information contained in the interblock comparisons, and hence, are comparatively less efficient. They have suggested the use of ranking after alignment and also considered the Wilcoxon's and Kruskal-Wallis tests based on aligned observations whose asymptotic efficiency have been studied very recently by Mehra and Sarangi [14]: (After the first draft of the paper was submitted, the author came to know through the editor about the paper by Mehra and Sarangi [14], submitted earlier to the Annals. In view of this, the overlapping part is not considered here.) The object of the present investigation is to formulate briefly the theory of rank order tests based on observations after alignment and through a multivariate approach to justify the validity and efficacy of the proposed tests (which include the earlier works as special cases).

We are motivated by Stein's proof (Stein (1946), Wald (1947), pages 157-158) of the termination of a sequential probability ratio test in the case of independent and identically distributed random variables. Extending his ideas to take certain "dependencies" into account we examine the rank-order sequential probability ratio test based on a Lehmann alternative studied in a paper with the above title by I. R. Savage and the author (1966) (referred to as SS I in the rest of this paper). We prove that this test terminates with probability one and that the stopping time has a finite moment generating function under a very mild condition on the bivariate random variables which resembles the Stein-condition, namely that a certain random variable $V(X_1, Y_1)$, defined in (32), is not identically equal to 0. Finally the asymptotic normality of the logarithm of the likelihood ratio of the rank order is established using the well-known Chernoff-Savage Theorem.

This paper attempts to add to the existing understanding of weighing designs (WD). Let us suppose that $n$ objects are to be weighed in $n$ weighings with a chemical balance having no bias. In the previous papers [14], [15], for odd $n$, optimum weighing designs are given for some particular cases subject to the conditions, viz., (i) variances of the estimated weights are equal and (ii) the estimated weights are equally correlated. Assuming these conditions we find some more optimum designs when $n$ is odd. Throughout this paper $n$ is assumed to be odd, except in Section 4, where $n$ may take any value. Let $x_{ij} = 1$ if the $j$th object is placed in the left pan in the $i$th weighing $= -1$ if the $j$th object is placed in the right pan in the $i$th weighing $= 0$ if the $j$th object is not weighed in the $i$th weighing. The $n$th order matrix $X = ((x_{ij}))$ is known as the design matrix. Also let $y_i$ be the result recorded in the $i$th weighing, $\epsilon_i$ is the error in the result, $w_j$ the true weight of the $j$th object, so that we have $n$ equations $x_{i1}w_1 + x_{i2}w_2 + \cdots + x_{in}w_n = y_i + \epsilon_i,\quad i = 1, 2, \cdots, n.$ We assume $X$ to be a non-singular matrix. The method of least squares or the theory of linear estimation gives the estimated weights $\hat{w}$ by the equation $\hat{w} = (X'X)^{-1}X'y$ where $\hat{w}$ and $y$ are the column vectors of the estimated weights and the observations respectively. If $\sigma^2$ is the variance of each weighing, then $V (\hat{w}) = (X'X)^{-1}\sigma^2 = ((c_{ij}))\sigma^2$ where $((c_{ij}))$ is the inverse of $X'X$. The design will be called optimum in the sense of Mood, if $\det ((c_{ij}))$ ($\det$ stands for determinant) is minimum and this is the case when $\det X'X$ is maximum. The efficiency of the weighing design can be measured, in the sense of Mood, by $\det X'X/\max \det X'X$ [12]. If $\lambda_{\min}$ be the minimum characteristic root of $X'X$, then the efficiency of the design can be measured, in the sense of Ehrenfeld, by $\lambda_{\min}/\max \lambda_{\min}$ [6]. According to Kishen, the efficiency can be measured by $1/\sum c_{ii}$ [11]. The first two definitions of efficiency were first introduced by Wald in [16]. Kiefer gave some optimality criteria for experimental designs in which the three criteria mentioned above were included and discussed them fully in [9] and [10]. He named these three definitions as $D$-optimum, $E$-optimum and $A$-optimum. We use Kiefer's notations in Section 6 for the definitions of efficiency of a weighing design given above.

It is desired to estimate $s$ parameters $\theta_1, \theta_2, \cdots, \theta_s.$ There is available a set of experiments which may be performed. The probability distribution of the data obtained from any of these experiments may depend on $\theta_1, \theta_2, \cdots, \theta_k, k \geqq s.$ One is permitted to select a design consisting of $n$ of these experiments to be performed independently. The repetition of experiments is permitted in the design. We shall show that, under mild conditions, locally optimal designs for large $n$ may be approximated by selecting a certain set of $r \leqq k + (k - 1) + \cdots + (k - s + 1)$ of the experiments available and by repeating each of these $r$ experiments in certain specified proportions. Examples are given illustrating how this result simplifies considerably the problem of obtaining optimal designs. The criterion of optimality that is employed is one that involves the use of Fisher's information matrix. For the case where it is desired to estimate one of the $k$ parameters, this criterion corresponds to minimizing the variance of the asymptotic distribution of the maximum likelihood estimate of that parameter. The result of this paper constitutes a generalization of a result of Elfving [1]. As in Elfving's paper, the results extend to the case where the cost depends on the experiment and the amount of money to be allocated on experimentation is determined instead of the sample size.

Let $W(t)$ denote a standard Wiener process for $0 \leqq t < \infty$. We compute the probability that $W(t) \geqq g(t)$ for some $t \geqq \tau > 0$ (or for some $t > 0$) for a certain class of functions $g(t)$, including functions which are $\sim (2t\log\log t)^{\frac{1}{2}}$ as $t\rightarrow\infty$. We also prove an invariance theorem which states that this probability is the limit as $m \rightarrow \infty$ of the probability that $S_n \geqq m^{\frac{1}{2}}g(n/m)$ for some $n \geqq \tau m$ (or for some $n \geqq 1)$, where $S_n$ is the $n$th partial sum of any sequence $x_1, x_2,\cdots$ of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. The main results were announced in [19]. Some aspects of the invariance theorem were considered independently by Muller [14], who also studied the rate of convergence to the limiting distribution. Statistical applications of these ideas are indicated in [3] and [18]. In Section 2 we state the general theorems and give several examples. Sections 3-5 are devoted to the proof of these results. In Section 6 we indicate the applicability of our methods to stochastic processes other than the Wiener process. Of particular interest in this regard is the analogue of Theorem 1 for Bessel diffusion processes. Section 7 raises questions which will be treated in a subsequent paper.

In this paper we shall consider mathematical models motivated by chemical process control problems in which we wish to control the process to hold some property of the output material as nearly constant as possible. For example, we might control the viscosity of the process output by varying the setting of a valve in the cooling water line to a heat exchanger. We shall suppose that the viscosity is observed at equally spaced times and that these observations form a sequence $Y_1, Y_2, Y_3, \cdots$. The control procedure specifies the valve setting $X_1$ to be used during the time interval $(0, 1\rbrack$. Then, immediately after observing $Y_n$, we reset the cooling water valve to the position $X_{n+1}$ in accordance with the control procedure. The value of $X_{n+1}$ may depend on the prior values $X_1, \cdots, X_n$ and $Y_1, \cdots, Y_n$. In choosing $X_{n+1}$ we hope to hold $Y_{n+1}$ as close to a fixed value $Y_0$ as possible. More specifically, our objective will be to keep values of the loss $E(Y_n - Y_0)^2$ small. In this paper it is assumed that time lags attributable to process dynamics are negligible when compared with the interval between observations. Parts of the theory are being generalized to allow for the presence of such time lags and no serious difficulties have as yet been encountered in this type of generalization. The basic process and the general control procedure are described in mathematical terms in Sections 2 and 3. The concept of a system consisting of the process and the control procedure is then introduced in Section 4. The specific type of control procedure we shall consider is a generalization of the simple proportional control procedure discussed by Box and Jenkins [1]. This procedure requires that we specify $X_1$ and then recursively specify $X_{n+1} = X_n - a_n(Y_n - Y_0)$. We may place bounds on $X_n$ as in (8) below. The sequence $\{a_n\}$ is a sequence of positive numbers which may be specified a priori as in Section 5 or sequentially as in Section 9. In Sections 6 and 7 we study the performance of certain processes when the sequence $\{a_n\}$ is specified a priori. Most important are cases where $a_n$ converges to a small value $a$ near zero. With the class of processes used in this paper, it turns out that this procedure for choosing $X_n$ is quite similar to the stochastic approximation procedure first presented by Robbins and Monro in 1951 [7] and later generalized by Dvoretzky [5] and other authors. A comprehensive review of work on stochastic approximation is contained in a recent article by Schmetterer [8]. In Section 8 we specialize to the study of certain stationary processes. Attention is focused on the asymptotic performance of the controlled process as a function of the limit $a$ of the sequence $\{a_n\}$ in our control procedure. Of particular interest is the asymptotic loss given by $\lim_{n\rightarrow\infty} E(Y_n - Y_0)^2$. A procedure for the sequential choice of the sequence $\{a_n\}$ is proposed in Section 9 and its theoretical properties are studied in Section 10. Conditions are given under which both $a_n$ and $E(Y_n - Y_0)^2$ converge. It turns out that this procedure for the sequential choice of $a_n$ is also a stochastic approximation procedure of the Robbins-Monro type. Numerical examples given in Section 11 indicate that the asymptotic loss obtained by using this procedure for sequentially choosing $a_n$ may be nearly as small as the asymptotic loss obtained when the sequence $\{a_n\}$ is chosen a priori to converge to the value of $a$ that minimizes the asymptotic loss. However, when $a_n$ is chosen sequentially, we need much less a priori knowledge of the process.

In a recent paper A. N. Shiryaev [1] discusses the problem of detecting the arrival of a "disorder" in an observed stochastic process, as quickly as possible subject to a limitation on the number of false alarms. He considers two versions of a simple model. In the first, the disorder arrives at a discrete instant $\theta$ according to a geometric distribution. The process disturbed by this event consists of a sequence of independent observations $\{\xi_t\}$, such that $\xi_1, \xi_2, \cdots, \xi_{\theta - 1}$ arise from a certain distribution $F_0$, whereas $\xi_\theta, \xi_{\theta + 1}, \cdots$ come from a different distribution $F_1$. In the continuous time version of the model, the a-priori distribution of $\theta$ is exponential: $P(\theta > t) = e^{-\lambda t} \quad (t \geqq 0)$ and the disorder is represented by a change in the mean drift of an observed Wiener process $\{\eta(t)\}$. More precisely, for any given value of $\theta$, this process has independent normal increments $\delta\eta = \eta(t + \delta t) - \eta(t)$, with \begin{align*}E (\delta\eta) = 0 \quad (0 \leqq t < \theta), \notag \\ E(\delta\eta) = \delta t \qquad (t \geqq \theta), \notag \\ \operatorname{Var} (\delta\eta) = \delta t \quad (t \geqq 0)\end{align*}. In both versions it may be decided at any instant $t$ to carry out a detailed inspection in order to ascertain whether or not the disturbance has occurred. Then, if it is found that $\theta < t$ the process terminates but observation must be resumed immediately after a false alarm. Within these rules it is required to find a decision procedure which determines the instants at which a thorough inspection is worthwhile. Assuming that $N$, the expected number of false alarms, is specified in advance Shiryaev establishes the general form of policy which minimizes $\tau$, the expected delay in verifying the arrival of the disorder. The a posteriori distribution of $\theta$ at any time, does not depend on anything which took place before the last false alarm. For example, in continuous time $p(t) = P(\theta \leqq t \mid \eta(t'), 0 \leqq t' \leqq t) = P(\theta \leqq t \mid \eta(t'), s \leqq t' \leqq t),$ where $s$ is the instant of the most recent false alarm. The geometric and exponential distributions have the useful property that $P(\theta > t + s \mid \theta > s) = P(\theta > t \mid \theta > 0).$ He deduces that the optimal policy for the period following any false alarm must correspond exactly with the procedure applied initially, before the first inspection. In addition, he proves the existence of a critical level $p^\ast = p^\ast(N)$ such that, in general, observations should continue so long as $0 \leqq p(t) < p^\ast$ with an immediate inspection as soon as $p(t) \geqq p^\ast$. These and other similar results are established first for the discrete time model and then extended to the continuous time version. For the latter, the paper also gives more explicit calculations: the evaluation of $\tau$ in terms of $p^\ast$, for example. But no attempt is made to determine the critical level $p^\ast(N)$ for the optimal policy. In fact, as we shall see, a very simple relation holds: $p^\ast (N) = (N + 1)^{-1}.$ However, the aim here is to show how the optimal policy can be found for a more realistic specification of the minimization problem, involving given delay and inspection costs. We shall concentrate entirely on the continuous time model and suppose that each inspection incurs an instantaneous cost $K > 0$, not depending on its outcome, whereas any delay in detecting the arrival of the disorder leads to a cost $c > 0$ per unit time. Hence the total expected cost is $K(N + 1) + c\tau$, which depends both on the decision procedure and on the initial condition $p(0) = 0$. The minimization will be based on the calculation of the minimum expected future cost $f_\ast(\varphi)$, as a function of the current state $\varphi(t) = p(t)/(1 - p(t))$, by solving a certain differential equation with special boundary conditions. A heuristic argument, in which one simply assumes that $f_\ast(\varphi)$ is suitably differentiable, can be given without much difficulty. But, strictly speaking, it is not clear that risk functions such as $f_\ast(\operatorname{var})$, each of whose values is defined as the infimum of a class of expectations, are sufficiently well behaved. This difficulty is often encountered in statistical applications of dynamic programming to processes in continuous time. Typically, it is extremely difficult to establish the required differentiability properties directly and it is necessary to seek an indirect justification by means of existence and uniqueness theorems. In our case the formal solution can be produced explicitly and, because of this, its justification is much easier. Nevertheless, the approach is complicated by the need to establish several preliminary results, and the discussion of these special properties is limited to a brief indication of the main steps, in the hope that the essential structure of the argument will be more generally useful. Section 2 is concerned with the information process $\{\varphi(t)\}$. Its relation to the observed process $\{\eta(t)\}$ is described and certain properties of its increments are collected for later use. The main argument begins in Section 3 with a discussion of sub-optimal decision procedures defined by specifying an open continuation set $\mathscr{C}$ within the space $\lbrack0, \infty)$ of possible "initial" states $\varphi$. It is shown that any risk function $f(\varphi)$ can be determined for each sub-interval of the corresponding set $\mathscr{C}$ by solving the basic differential equation appropriately. Sections 4 and 5 consider the special solution $f_\ast(\varphi)$ which represents the optimal decision procedure, and give the required verification that $f_\ast(\varphi)$ is uniformly minimal. The final section contains a brief analysis of the operating characteristics of the optimal policy and indicates the importance of evaluating the particular minimum risk $f_\ast(0)$.

The subject of minimum variance unbiased estimation has received a great deal of attention in the statistical literature, e.g., in the papers of Bahadur [2], Barankin [3], and Stein [14]. The emphasis in these papers has typically been placed on the existence and construction of minimum variance unbiased estimators when the sampling plan to be used was given in advance. In this paper, criteria are developed for the selection of an appropriate sampling plan for the family of binomial distributions. Thus, independent observations are to be taken on the random variable $U$ so distributed that \begin{equation*}\tag{1.1} \operatorname{Pr} \{U = 1 | p\} = p,\quad \operatorname{Pr} \{U = 0 | p\} = 1 - p = q,\end{equation*} where $p$ lies in the open interval $0 < p < 1$, and the value of a given function, $g(p)$, is to be estimated. The problem considered here is that of determining a sampling plan and an unbiased estimator of $g(p)$ that are optimal, in some sense, at a specified value, $p_0$, of $p$. Optimality will depend, not only on the variance of the estimator, but also on the average sample size of the plan. A sampling plan, $S$, and estimator, $f$, will be considered optimal at $p_0$ if, among all procedures with average sample size at $p_0$ no larger than that of $S$, there does not exist an unbiased estimator with smaller variance at $p_0$ than that of $f$. The basic tool to be used is the information inequality (see Lemma 2.7 and the discussion following it) which provides a lower bound for the variance of an estimator in terms of its expected value and the average sample size of the sampling plan. If, at $p_0$, this lower bound is attained for a particular estimator and sampling plan, it may be immediately concluded that they are optimal at $p_0$. Such an estimator is said to be efficient at $p_0$. In Section 2, various definitions, assumptions, and fundamental facts to be used throughout the paper are collected. In Section 3 it is shown that the single sample plans and the inverse binomial sampling plans are the only ones that admit an estimator that is efficient at all values of $p$. In Section 4 some techniques are given that are often useful in the analysis of inverse binomial sampling plans. In Section 5 relationships between the average sample size of a sampling plan and the functions of $p$ that are estimable optimally are explored. In Sections 6 and 7 it is shown that there can exist two distinct sampling plans with the same average sample size for all $p$ and some comparisons are made of such plans. In Section 8 a new characterization of completeness is given for bounded sampling plans and it is shown that the dimension of the linear space of unbiased estimators of 0 can be determined simply by counting the number of boundary points. It is further shown that for a wide class of plans, the estimators that are efficient at a given value of $p$ do not have uniformly minimum variance, although non-trivial uniformly minimum variance estimators do exist. After this paper had been accepted for publication I learned that R. B. Dawson had obtained expression (4.13) and Theorems 8.2 and 8.4, as well as various other interesting results related to the material of Section 8, in his Ph.D. thesis, "Unbiased tests, unbiased estimators, and randomized similar regions," Harvard University, May, 1953.

It may frequently happen that a researcher, wishing to decide which one of a set of alternatives to accept, finds that there are several experiments available to him which he might perform to guide him in reaching his decision. Thus he is faced with making a preliminary decision as to which experiment or experiments he is to perform. If he admits the possibility of performing more than one, then the question of how many, which ones, and in what order arises. It is such questions as these that come under the heading of comparison and design of experiments. While a great deal of general theory of the design problem has been developed, e.g., by Wald [1] and Maguire [2], few actual solutions of particular problems, especially of the sequential type, have been investigated thus far. Robbin's paper [3] is the first published report dealing with various sequential rules for particular nontruncated design problems. The basic purpose of this paper is to investigate for certain cases the optimal design, which almost uniformly turns out to be exceedingly complicated (see Section 3), and to propose and determine some justification for certain simpler criteria. Attention is restricted to problems in which there are but two alternatives, or hypotheses, $H_1$ and $H_2$, and it is required to decide between them with a loss of one unit if the false one is selected, while no loss results from selecting the correct one. Further, $\xi$ will denote the a priori probability that $H_1$ is true, and the basic criterion for comparison will be the Bayes risks associated with the various experiments. To say that an experiment is available to a researcher is to say that there is a real random variable which he can observe whose distribution is known under each hypothesis. As an example of a situation in which this type of question may arise, consider the problem of deciding between utilizing a use test as against a specifications test for acceptance of a lot of manufactured items. A large lot of items has been produced and a decision is to be made between, say, $w_1$ and $w_2$ as being the proportion of defectives in the lot. Let $X = 1$ or 0 according as an item selected at random is defective or not as determined by subjecting it to a use test. Let $Y = 1$ or 0 according as an item selected at random is classified as defective (because it fails to meet certain specifications) or not. If $\alpha$, the probability that a nondefective item fails to meet the specifications, and $\beta$, the probability that a defective item meets the specifications, are known, then both $X$ and $Y$ have a binomial distribution with known parameter under each hypothesis. Again, it might be that in the course of a series of treatments of a material there are two points at which a certain characteristic may be measured, say the breaking strength of a metal undergoing a series of heat treatments. Let $X$ and $Y$, respectively, denote the value of the characteristic at the different points in the process. It is reasonable to assume that under each of two simple hypotheses concerning the process and material, $X$ and $Y$ have prescribed normal distributions. In general, suppose that $X$ and $Y$ are two real random variables having distribution functions of $F_i$ and $G_i$, respectively, under hypothesis $H_i$ and with corresponding densities $f_i$ and $g_i$ with respect to a common measure, $\Psi$, such that $f_i > 0$ if and only if $g_i > 0$. Let $R_Z(\xi)$ denote the Bayes risk against $\xi$ when using experiment $Z$. Now the computation and comparison of the risk, $R_X(\xi)$ and $R_Y(\xi)$, for all $\xi$, which appears to be necessary in order to obtain an optimal design, is intrinsically complicated in most cases, as will be seen in the following sections. Hence it is of some interest to investigate some more convenient criteria for choosing between experiments. Any such criterion should, of course, dictate the use of $X$ if $R_X \leqq R_Y$ (i.e., $R_X(\xi) \leqq R_Y(\xi)$ for all $\xi$). To be able to check on this is but one reason for interest in conditions that $R_X \leqq R_Y$. Another is that whenever it is true that $R_X \leqq R_Y$, then regardless of the choice of actions open to the researcher or the loss function used, use of $X$ will never yield a greater Bayes risk than $Y$. Also, if a total of, say, $n$ independent experiments is to be performed, the optimal sequential design is the nonsequential rule: take all $n$ observations of $X$ ([4], [5], [8]). In Section 2.1 general conditions that $R_X \leqq R_Y$ are derived, some of which are related to those obtained by Blackwell [5] via consideration of the standard experiment. In Section 2.2 the Kullbach-Leibler (abbreviated hereafter as K-L) information numbers are introduced, and it is shown, in particular, that they provide a criterion which yields an especially simple necessary condition that $R_X \leqq R_Y$. The K-L information numbers are also considered as functions of that transformation, $t$, such that the distribution of $t(X)$ under $H_1$ is the distribution of $X$ under $H_2$. The case in which all the distributions involved are normal is analyzed in some detail in Section 2.3, where it is seen that the K-L numbers do not yield a sufficient condition that $R_X \leqq R_Y$, though they do yield a sufficient condition that $R_X = R_Y$. The normal case gives an example in which a second criterion, that of being "locally more informative" (Bayes) at both zero and 1, yields a condition both necessary and sufficient that $R_X \leqq R_Y. X$ is termed locally more informative than $Y$ at $\xi$ if $\xi$ lies in an interval $\lbrack\xi_1, \xi_2\rbrack$ such that on $\lbrack 0,1\rbrack \cap \lbrack\xi_1, \xi_2\rbrack R_X(\xi) \leqq R_Y(\xi)$, with strict inequality at at least one end point. This latter criterion is discussed further in Section 2.4. The problem of determining the optimal designs, sequential and nonsequential, for the case in which all distributions are binomial and a fixed number of experiments is to be performed is discussed in Section 3. The complete solutions are found to be exceedingly complicated; a few are given. For the sequential design, a system for obtaining the optimal design which avoids the complete calculation of the successive risk functions was found. In the final section, a sequential rule for terminating experimentation is considered, and the problem of finding a sequential design which minimizes the expected number of experiments is posed. Two reasonable designs are proposed, are shown to be equivalent, and are shown to be better than either of the rules which require that all observations be of the same random variable. Throughout the paper it will be convenient to consider $\xi/(1 - \xi)$, and this will regularly be denoted by $\eta$.

A queueing system can be regarded as transforming one point process into another (as pointed out for example in Kendall (1964), Section 6), namely, the input or arrival process with inter-arrival intervals $\{T_n\}$ is acted on by a system comprised of a queue discipline and a service (or, delay) mechanism, producing the output or departure process with inter-departure intervals $\{D_n\}$. The object of this paper is to study the correlation structure of the sequence $\{D_n\}$ (and this sequence we shall for convenience call the output process of the system) when the input process is a renewal process and when the service times $\{S_n\}$ (assumed to be independently and identically distributed, and independent of the input process) are such that the system can and does exist in its stationary state. In particular, we shall be concerned with conditions under which the process $\{D_n\}$ is uncorrelated, by which we mean that $\operatorname{cov} (D_0, D_n) = E(D_0D_n) - (E(D_0))^2 = 0 (n = 1,2, \cdots)$. Schematically then, we study the mapping $\{T_n\} \overset{\{ S_n\}/1}{\longrightarrow} \{D_n\}$, and as consequences of the formal theorems of the paper the following statements can be justified $(T, S$, and $D$ denote typical members of $\{T_n\}, \{S_n\}$ and $\{D_n\})$. (i) $\operatorname{var} (D) \geqq \operatorname{var} (S)$, with equality only in the trivial case where both $\{T_n\}$ and $\{S_n\}$ are deterministic. (ii) Locally, the mapping can be made any of variance increasing, variance preserving, or variance decreasing (that is, all cases of $\operatorname{var} (D) >, =, < \operatorname{var} (T)$ are possible) by appropriate choice of $\{T_n\}$ and $\{S_n\}$. Globally however, the mapping is variance preserving, that is, $\operatorname{var} (D_1 + \cdots + D_n)/\operatorname{var} (T_1 + \cdots + T_n) \rightarrow 1\quad (n \rightarrow \infty)$. (iii) When $\{T_n\}$ is a Poisson process, the process $\{D_n\}$ is uncorrelated if and only if it is a Poisson process (and this occurs if and only if the $\{S_n\}$ are negative exponential). (iv) When the $\{S_n\}$ are negative exponential, $\{D_n\}$ is a renewal process if and only if it is a Poisson process (and this occurs if and only if $\{T_n\}$ is a Poisson process). However (cf. (iii)) $\{D_n\}$ can be uncorrelated without being a renewal process. If the $\{D_n\}$ are correlated then the terms $\operatorname{cov} (D_0, D_n)$ are of the same sign for all $n = 1,2, \cdots$ and converge to zero monotonically. (v) There exist $\{T_n\}$ and $\{S_n\}$ such that the serial covariances $\operatorname{cov} (D_0, D_n)$ are not of the same sign for all $n = 1,2, \cdots$ (see remark after Theorem 7).