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The article presents abstracts and titles of abstracts submitted for papers presented at the 1969 Annual Meetings of the American Statistical Association and Biometric Society in New York. Some of the abstracts are Financial and Non-Financial Factors Affecting Post-High School Plans, 1939-1965," by Walter Adams, "Experimental Designs and Mathematical Models," by Sidney Addelman, "Consistency of Linear Least Squares Estimate in a Regression Model with Lagged Variable," by M. Ashanullah, "The Use of Prior Information in Testing of New Drugs," by David M. Allen, "A Comparison of Probability Density Estimates," by Gary Anderson, "Stochastic Models of City Population Density," by Robert T. Amsden, "ldentification and Input Signal Synthesis Problems in Discrete-Time Stochastic Control Systems," by Masanao Akoi, "A Computer Language for the Analysis of Variance," by David J. Aemor, "A Generalized Mathematical Model for Biological Recovery with Applications in Radiation Biology," by George M. Angelton."

We want to do three things in this paper: (1) to propose a more general probability model than the one considered by Toby or Davies for dealing with the number of buses a student would see, on the average, going in the wrong direction; (2) to show that, with this general model, one would not see more wrong-direction buses, or the same number of wrong-direction buses, as suggested by Toby and Davies, but fewer wrong-direction buses; (8) to question the advisability of using a student's experiences with wrong-direction buses as the way to shake his faith in the validity of personal experience. As Davies has said, clarification is especially necessary now that Toby's paper has appeared in books of readings designed for the course in introductory sociology. [ABSTRACT FROM AUTHOR]

The lack of memory property of the exponential distribution plays an important part in the branch of applied probability. This property assumes the information regarding the probability distribution. In the present paper we give a characteristic property of the exponential distribution based on the means of the conditional distributions. Considering a random variable T with finite mean and such that P(T > O) > 0, and denoting by y a positive number such that P(T > y) > 0, we show that T has an exponential distribution if and only if the mean of the conditional distribution, given T > y, exceeds the mean of the unconditional distribution by the quantity y for all y. Also given in the paper is a similar characterization for the geometric distribution. Since the information concerning the expected values is easily accessible, we expect these properties to be useful in dealing with the practical problems. [ABSTRACT FROM AUTHOR]

The model considered in this paper is simple linear regression (Ey[sub i] = beta[sub 1] + beta[sub 2] x[sub 1], i = 1, ..., n), and the problem is to make statistical inferences about an unknown value of x corresponding to one or more additional observed values of y. The maximum likelihood estimator x of x and the classical (l - alpha) 100% confidence set S for x have some undesirable properties. For example, x has infinite mean square error and P {S = (- infinity, + infinity)} > 0. The purpose of this paper is to demonstrate that insight and understanding, as well as a useful class of solutions, can be obtained by looking at the problem from a Bayesian point of view. A result which follows from a general Bayes solution is that the inverse estimator [4] is Bayes with respect to a particular informative prior. [ABSTRACT FROM AUTHOR]

The article discusses moments of the distribution of sample size, N in a sequential probability ratio test (SPRT). The present paper provides variance, the third and the fourth moments of N. The details are worked out in five common applications of the SPRIT. The relation of the variance of N to the truncation of a SPRT is discussed is also discussed in the paper. Scholar A. Wald indicated in passing how one can obtain the moments of N, but the only published literature where the author encountered a general expression for the variance of N. However, their expression is incorrect. Using scholar J. Wolfowitz's results, which they do, or differentiating Wald's, fundamental identity twice one gets provided. In many practical applications of the SPRT, μ and moments in an equation derived are differentiable functions of a real-valued parameter. The limiting expressions for the moments can then be determined by standard methods of mathematical analysis. However, for the third and fourth moments the actual technique may involve an excessive amount of arithmetic.

In an earlier paper [5] the authors compared the power of the BLUS test with the probability of a correct decision of the Durbin-Watson bounds test. A method to compute the distribution of the Von Neumann ratio under the null hypothesis and under the alternative hypothesis was given. In the present paper the latter method is used to tabulate the BLUS-test statistic and to compute the exact significance points of the Durbin-Watson test for several examples. Powers of both tests are computed and compared. It appears that, for the cases considered, the power of the exact Durbin-Watson test exceeds that of the BLUS procedure, while the latter is greater than the probability of a correct decision in the Durbin-Watson bounds test. [ABSTRACT FROM AUTHOR]

This paper concerns a new Normal approximation to the beta distribution and its relatives, in particular, the binomial, Pascal, negative binomial, F, t, Poisson, gamma, and chi square distributions. The approximate Normal deviates are expressible in terms of algebraic functions and logarithms, but for desk calculation it is preferable in most cases to use an equivalent expression in terms of a function specially tabulated here. Graphs of the error arc provided. They show that the approximation is good even in the extreme tails except for beta distributions which are J or U shaped or nearly so, and they permit correction to obtain still more accuracy. For use beyond the range of the graphs, some standard recursive relations and some classical continued fractions are listed, with some properties of the latter which seem to be partly new. Various Normal approximations are compared, with further graphs. The new approximation appears far more accurate than the others. Everything an ordinary user of the approximation might want to know is included in this paper. The theory behind the approximation and most proofs are postponed to a second paper immediately following this one. [ABSTRACT FROM AUTHOR]

In this paper a method is developed for estimating the parameters in the multivariate normal distribution in which the missing observations are not restricted to follow certain patterns as in most previous papers. The large sample properties of the estimators are discussed. Equivalence with maximum likelihood estimators has been established for a subclass of problems. The results of some simulation studies are provided to support the theoretical development. [ABSTRACT FROM AUTHOR]

The 1967 Committee on Publications, chaired by David L. Wallace, found that many American Statistical Association members desired more review and survey papers. These have been hard to come by and so as the author's last act before leaving office, decided to provide a short survey paper on some related ideas in a field where nearly all of the statisticians sometimes work — that of contingency tables. These ideas are largely available in literature and yet they have not often been put together, though statisticians I. J. Good's monograph and Leo Goodman's many papers form good sources. But the author's paper is not intended as a review of the literature, only as a survey of one set of ideas about estimation in the analysis of contingency tables. The author fears that the first act of most social scientists upon seeing a contingency table is to compute chi-square for it. Sometimes this process is enlightening, sometimes wasteful, but sometimes it does not go quite far enough. The author collected 500 samples of writing published about 1961.

The Bradley-Terry model for a paired-comparison experiment with t treatments postulates a set of t 'true' treatment ratings π1, π2, ..., π1 such that π1 is ≥ 0, σpi;i = 1 and the probability for preferring treatment i to treatment j is π2 (π2 + πj)-1. Thus, according to this model, every comparison of two treatments results in a definite preference for one of the two. This is an unrealistic restriction since when there is no difference between the responses due to two treatments, any method of expressing preference for one over the other is somewhat arbitrary. This paper considers a modification of the Bradley-Terry model by introducing an additional parameter, called threshold parameter, into the model. This permits 'ties' in the model. The problem of estimation and tests of hypotheses for the parameters of the modified model is also dealt with in the paper. [ABSTRACT FROM AUTHOR]

An earlier paper [3] provides axioms for the seasonal adjustment of economic time series. Seasonally adjusted data should be minimum variance, unbiased, and linear in an appropriate sense. These axioms yield a unique method for seasonal adjustment. The seasonally adjusted data may be obtained by an application of ordinary least squares regression. The problem of seasonal adjustment of economic time series in [3] is not the same as the problem of seasonal adjustment of data for econometric analysis. The first problem is completely resolved by appeal to the axioms of minimum variance, unbiasedness, and linearity. The second problem requires formulation as a standard problem in econometrics: estimation of the parameters of a single equation in a system of simultaneous equations. Lovell [4] has proposed to solve the problem of seasonal adjustment of data for econometric analysis by applying ordinary least squares directly to a structural equation in a system of simultaneous equations. Ordinary least squares estimation of the parameters of a structural equation usually results in "least squares bias." However, under certain special assumptions such a procedure can be justified, as Wold [6] has pointed out. Under these assumptions Lovell's procedure is valid. In this paper the seasonal adjustment of data for econometric analysis is formulated as a general simultaneous equations problem. Conditions for identification of the parameters to be estimated and methods for constructing consistent and asymptotically unbiased and efficient estimates are derived. Extensions to multivariate problems of seasonal adjustment for econometric analysis are sketched. [ABSTRACT FROM AUTHOR]

Given N points independently drawn from the uniform distribution on (0, 1), let p[sub n], be the size of the smallest interval that contains n out of the N points; let n[sub p], be the largest number of points to be found in any subinterval of (0, 1) of length p. This paper uses a result of Karlin, McGregor, Barton and Mallows to determine the distribution of n[sub p] for p = 1/k, k an integer. The paper gives simple determinations for the expectations and variances of p[sub n], for all fixed n > (N + 1)/2, and of n[sub 1/2]. The distribution and expectation of n[sub p] are estimated and tabulated for the cases p = 0.1(0.1)0.9, N =2(1)10. [ABSTRACT FROM AUTHOR]

Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability for the ith unit to be in the sample is proportional to its 'size' x. From the alternative methods of achieving this we consider here only the so-called systematic method which, to the best of our knowledge, was first developed by W. G. Madow (1949): The units in the population are listed in a 'particular' order, their x, accumulated and a systematic selection of n elements from a 'random start' is then made on the accumulation. In a more recent paper (H. O. Hartley and J. N. K. Rao (1962) ) an asymptotic estimation theory (for large N) associated with this procedure was developed for the case when the order of the listed units is random. In this paper we draw attention to certain properties of Madow's estimator: We utilize the fact that with systematic sampling the total number of different samples is N (rather than ([This eq. cannot be change in char.]) as with completely random sampling). This simplification in the definition of the variance of the estimator in repeated sampling enables us to identify the exact variance of Madow's estimator with a 'between sample mean square' in a special analysis of variance (see section 4) and compare it with the variance of the pps estimator in sampling with replacement as well as in other sampling procedures. We also develop two approximate methods of variance estimation (see section 5). We pay particular attention to the case when the units are listed in the order of their size. With this particular arrangement our method can be described as 'systematic with random start' and the gain in precision that we accomplish has of course, analogues in systematic sampling with equal probabilities employing ratio estimators in which there is a relation between the ratio ri =yi/Xi and xi Compared with other methods the present procedure combines the advantage of ease of systematic sample selection with the availability of exact variance formulas for any n and N. Moreover, it usually leads to a more efficient estimate. Its shortcoming resides in the fact that the estimation of the variance is based on certain assumptions. [ABSTRACT FROM AUTHOR]

Various methods based on Student t variates have been suggested and used for obtaining simultaneous confidence intervals for several means, or for several contrasts among means. Determination of an overall confidence level for such intervals involves evaluating the probability mass of a multivariate t distribution over a hypercube centered at the origin, with sides paralleling the coordinate planes, or obtaining bounds for this probability mass. Since such distributions involve many nuisance parameters, an impossible number of tables would be necessary in order to make exact confidence intervals. In the virtual absence of tables, approximations and bounds become important. In this paper, an attempt has been made to investigate the adequacy of certain suggested approximations [2], [5], [8] by computing the exact distributions for some particular cases. These exact distributions have been compared with approximations. This paper is concerned with two-sided confidence intervals, rather than one-sided intervals. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic relative efficiency (A.R.E.) and the asymptotic power of certain tests. These tests are distribution-free goodness-of-fit tests based on the empirical distribution function. They are the one and two sided Kolmogorov-Smirnov tests and a new one based on the absolute values of the order statistics. The method of obtaining the A.R.E. is due to Quade [5] and is outlined in Section 1. In this paper we are concerned with the case of scalar alternatives. The asymptotic power for the Kolmogorov-Smirnov tests is obtained in Section 2. Section 3 deals with a new test which allows a more facile use of a one-sided statistic while utilizing the same information normally included in a two-sided statistic. In Section 4, three examples are considered to illustrate the superiority of this new test. [ABSTRACT FROM AUTHOR]

The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the vivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio (u[sub 1] + ... + u[sub n])/(v[sub I] + ... + v[sub m]) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed. [ABSTRACT FROM AUTHOR]

The use of simulation, as a technique for attacking difficult problems, has increased greatly with the availability of the digital computer. This is illustrated by the large number of references in Shubik's (1960) bibliography[sup 2] and in the large number of studies published since then. Simulation is essentially an extension of a technique known as empirical sampling, or distribution sampling, which has been used in the field of statistics for many years. The limitations of the technique, which are well known to statisticians, are apparently not as well known, or at least not as well recognized, by those using simulation today. The first part of this paper contains an historical survey of distribution sampling as used by statisticians. The material was originally prepared in 1953 and is reproduced here in slightly revised form to bring this history to the attention of present day simulators in order that the lessons that can be learned from this part can more readily be incorporated in the development of methodology today. The second part of this paper discusses the relevance of empirical sampling to the present day state of the art of simulation. The technique of generating random members, developed for empirical sampling can be applied directly to simulation. However in other aspects simulation is more difficult than empirical sampling and here the theory of distribution sampling does not have much to offer. The difficulties are due to lack of independence among time series, non-stationarity of the time series, and the large number of parameters. [ABSTRACT FROM AUTHOR]

This paper examines Caplow's model of coalitions in the triad and identifies several shortcomings. Retaining the basic structure of Caplow's arguments, it recasts the problem as one of interdependent rational decision making. The reformulation allows increased generality in several directions, including a more general notion of strength, relaxation of the additivity assumption about strength, and the adoption of probabilistic choice to allow predictions about the relative frequencies of the various coalitions. [ABSTRACT FROM AUTHOR]

In 1965 Stanley Warner [8] illustrated a technique whereby one could estimate from a sample the proportion of persons in a population possessing some characteristic X, without pointedly asking the question, "Do you possess characteristic X?". The present article uses a variation of these ideas suggested by Warner in a later paper [7] to estimate the distribution function of a continuous-type random variable. The technique is illustrated by estimating an income distribution from the responses of a sample of 500 individuals. The potential use of devices of this type in maintaining confidentiality of existing data files is apparent. [ABSTRACT FROM AUTHOR]

Given N points distributed at random on [0,1), let np, be the size of the largest number of points clustered within an interval of length p. Previous work finds Pr (np ≥ n), for n> N/2, and for n < N/2, p=1/L, L an integer. The formula for the case p=1/L is in terms of the sum of L × L determinants and is not computationally feasible for large L. The present paper derives such a computational formula. [ABSTRACT FROM AUTHOR]

The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic pay. off function and a lower estimate with a piecewise linear payoff function, for example. [ABSTRACT FROM AUTHOR]

The article describes a process for determining migration probability. A stochastic process may be defined as a process which develops over time and where events at time 2 are not independent of events at time 1. This lack of independence may be viewed in probabilistic terms. Since the application of stochastic probability processes to problems of predicting individual behavior requires data obtained from case studies where each individual can be classified with respect to a given set of categories at successive points in time, the projected analysis required a set of individual longitudinal migration histories. Consequences of the basic mechanism of the model utilized in the paper are also detailed.

The intent of this paper is to provide a definition of a sociometric clique in the language of graph theory. This problem is viewed from two perspectives: maintaining fidelity to the intuitive notion of a clique; and providing adequate computational mechanics for large bodies of data. Luce's (1950) concept of an n-clique is used, but further qualifications are added. Two statistics or measures with associated probability distributions are defined for testing the adequacy of a subgraph which qualifies according to the definition. [ABSTRACT FROM AUTHOR]

This paper proposes several methods of estimating parameters in stable distributions. All the methods involve sample characteristic functions. One of the methods which is based upon the method of moments is treated in some detail. Asymptotic normal distributions for the proposed moment estimators are provided. Moreover, all methods provide consistent estimators. The estimation problem is treated for both univariate and multivariate stable distributions. [ABSTRACT FROM AUTHOR]

The article presents information on a nomogram for the "Student"-fisher t test. A nomogram is given for estimating the probability (P) for a given value of the "Student"-Fisher t test. W.S. Gosset, an employee of the Guiness brewing company in Dublin, published papers in 1908 in which he correctly solved three problems: the probable error of a mean, the distribution of the mean divided by its estimated standard deviation and the distribution of the estimated correlation coefficient between independent variates. Later "Student" and economist R.A. Fisher calculated tables of the relevant t distribution and Fisher gives a table of t and probabilities, corresponding to various degrees of freedom. Fisher and F. Yates, scholar provide in addition a column for P. It seemed that presentation of the P, degrees of freedom, t relationship in the form of a nomogram would be advantageous. It makes possible a fairly exact estimate of probabilities less than 0.0001 and makes it possible to get an estimate of P for any value of t from 1 to 65, instead merely of selected values.

In this paper methods for obtaining approximations and bounds for the moments of order statistics from a continuous parent distribution are discussed. These bounds and approximations depend on the distribution function only through certain moments of order statistics in small samples. It is shown that for the Cauchy distribution bounds and approximations of all finite moments can be obtained. Some numerical calculations for normal and Cauchy distributions are also given. [ABSTRACT FROM AUTHOR]

This paper uses a family of bivariate extreme-value distributions to estimate the probability of a large exceedance of a random variable given that a certain other random variable not independent of the first has exceeded a certain value. A simple method of reasonably good efficiency is given for estimating a bivariate extreme-value distribution from independent bivariate samples. The method is used to analyze the performance of a spacecraft command receiver which has an indication of data quality so that commands likely to be in error can be rejected. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to illustrate a "give and take" type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observations x and the unknown "true" state of nature sigma. Quite often, replication of the experiments reveals considerable variation in x, indicating that no mathematical treatment of the problem is possible without the assumption that x is a sample value of a random variable X, treated in terms of frequentist theory of probability. As to the true state of nature, sigma, situations vary. Indeed, there are cases where it appears natural to consider that sigma is selected at random out of a certain known set SIGMA, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can "give" something to science. In each example, however, the assumption of the randomness of sigma appears extraneous. Section 3 describes another example of research in science, again with a non-random sigma, which happened to "give" something to the frequentist theory of statistics. The motivation to reduce the frequency of erroneous conclusions in the general circumstances of the biological study had led to considering the possibility that a certain derivative may fail to exist. This created a problem of testing (Section 4) which does not appear to have been considered. A solution of the problem, in the form of optimal symmetric (C alpha) tests, is given in Section 5. [ABSTRACT FROM AUTHOR]

Maximum likelihood and other BAN estimators have been shown to possess certain optimal asymptotic properties in estimating the parameters of probability distributions satisfying specific regularity conditions. The subject of non-regular estimation is concerned with problems in which these conditions do not hold. In many such problems, classical-lower bounds on the variance of unbiased estimators, such as the Cramer-Rao bound, lead to the trivial result V(t) * (These characters cannot be converted in ASCII text) 0, where t is any unbiased estimator. A number of alternative bounds for application in the non-regular case have been derived. In this paper previous results of this type are reviewed and an additional bound is given. The specific applications of interest involve estimation of a location parameter. Applications of the bounds to the exponential, uniform and Pearson Type III distributions are investigated. [ABSTRACT FROM AUTHOR]

In the early years of the eighteenth century A. De Moivre, N. Bernoulli, and P. R. Montmort found three solutions for the following problem of games of chance: Two players, A and B, play a series of games. In each game, independently of the others, either A wins a counter from B with probability p or B wins a counter from A with probability q (p + q = 1). The series ends if either A wins a total number of a counters from B or B wins a total number of b counters from A. What is the probability that A wins the series in at most n games? Denote this probability by P[sub n](a,b). In this paper simple and elementary proofs are given for the various formulas for P[sub n](a,b). Furthermore, it is shown how these formulas can be applied in the theories of order statistics, random walks, storage, queues, Brownian motion, and dams. [ABSTRACT FROM AUTHOR]

Let X[sub (1)] is less than or equal to X[sub (2] is less than or equal to ... is less than or equal to X[sub (n)] be an ordered sample of a random variable X which has continuous probability distribution function F (x), and let F[sub n] (x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Renyi, are considered: [Multiple line equation(s) cannot be represented in ASCII text] The paper presents table of exact probabilities for these statistics for finite sample sizes. The limiting distributions of these statistics for sample size n arrow right Infinity are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions. Numerical examples for the use of the tables are presented, as well as applications to testing hypotheses on life distributions and to one-sided estimation of probability distribution functions from censored data. [ABSTRACT FROM AUTHOR]

Tests for randomness are described, when the data consists of points on the circumference of a unit circle, and the alternative to the uniform distribution (randomness) is either a unimodal or a bimodal distribution: the yon Mises distribution or an adaptation. Tables of significance points are given for the test statistics, and the power is used to give a table of sample sizes needed to detect a given degree of clustering, measured by the parameters of the distributions. This paper is a close parallel, for two dimensions, to Stephens [15], which gave corresponding tests and tables for three dimensions. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance-component estimation for the unbalanced two-way completely-random classification where the numbers of observations in the subclasses are treated as random variables not necessarily independent of some of the random effects of the model. General results are given on the expectations of two commonly-used estimators of the vector of variance components. Numerical approximations are presented for these expectations for one sub-family of the family of all possible joint distributions of the subclass numbers and the random effects. [ABSTRACT FROM AUTHOR]

The aim of this paper is two-fold: (1) To give a unified treatment of the theory of order statistics when the parent distribution is not necessarily continuous. (2) To assess the effects of grouping on the distribution of order statistics and to indicate the convenience, under suitable conditions, of using order statistics for the estimation of parameters from grouped data with or without censoring. [ABSTRACT FROM AUTHOR]

This paper describes and derives the asymptotic behavior of the new Normal approximation introduced in Part I and of a family of Normal approximations based on roots, including the square root approximations of Fisher, and Freeman and Tukey, and the cube root approximations of Wilson and Hilferty, Camp, and Paulson. Various asymptotic comparisons are made, all of which rank the new approximation first, the cube root approximations second, and the other root approximations (and the ordinary Normal approximation) third. For instance, in the binomial case, if the tail probability is fixed as n arrow right infinity, the errors resulting from the foregoing approximations are generally of order n[sup -3/2], n[sup -1], and n[sup -1/2] respectively, while for a tail probability approaching 0, the relative error in approximating it approaches 0 if the corresponding standard Normal deviate is of smaller order than n[sup 1/2] n[sup 1/4], or n[sup 1/6] respectively, but not generally otherwise. These comparisons are deduced from far more detailed results which are also given. [ABSTRACT FROM AUTHOR]

Subsets of the dependent variable of a linear model often conform to known constraints; for example, this situation arises with Engel curves and with models which estimate transition probabilities, market shares, or other classes of proportions. These models imply a variety of restrictions on the parameters, explanatory variables and residuals which must be treated explicitly in estimating and testing hypotheses about the parameters. This paper discusses these restrictions and develops an estimating procedure which yields consistent estimates which are asymptotically efficient, unbiased and normal. [ABSTRACT FROM AUTHOR]

In this paper, maximum likelihood and Bayesian methods are presented for estimating transition probabilities when data in the form of aggregated proportions are available. The probability function for the observed proportions is assumed to have a multinomial distribution under the Lexis scheme. The multivariate beta distribution is used as the prior probability density function in formulating the Bayesian estimator. The results of some Monte Carlo experiments provide some evidence on the sampling properties of several alternative estimators. [ABSTRACT FROM AUTHOR]

This paper gives analytic expressions for the distribution function of t", a doubly noncentral t-distribution, with nu degrees of freedom and noncentrality parameters delta and lambda, in series of Incomplete Beta and Hypergeometric (Gauss and Confluent) functions. Numerical values of Pr(t" ≤ t|v, δ, λ) for v = 2 (1) 20, λ = 0 (2) 8, |δ| = 0 (1) 5, t=0 and 1 are given in Table I. [ABSTRACT FROM AUTHOR]

Let X[sub i1], X[sub i2],..., X[sub iota n], I=1, 2, be a pair of random samples from populations which are normally distributed with means theta[sub iota], and common known variance tau[sup 2]. The problem is to estimate the function psi(theta[sub 1], theta[sub 2]) = maximum (theta[sub 1], theta[sub 2]). In this paper we consider five different estimators (or sets of estimators) for psi(theta[sub 1], theta[sub 2]) and evaluate their biases and mean square errors. The estimators are (I) psi(X[sub 1], X[sub 2]), where X[sub I] is the sample mean of the ith sample; (ii) the analogue of the Pitman estimator, i.e. the a posteriori expected value of psi(theta[sub 1], theta[sub 2]) when the generalized prior distribution is the uniform distribution on two dimensional space; (iii) a class of estimators which are generalized Bayes with respect to generalized priors which are products of uniform and normal priors; (iv) hybrid estimators, i.e. those which estimate by (X[sub 1] + X[sub 2])/2 when |X[sub 1] -X[sub 2]| is small, and estimate by psi(X[sub 1], X[sub 2]) when |X[sub 1] - X[sub 2]| is large; (v) maximum likelihood estimator. The bias and mean square errors for these estimators are tabled, graphed, and compared. Also the invariance properties of these estimators are discussed with a rationale for restricting to invariant estimators. [ABSTRACT FROM AUTHOR]

This paper presents three upper bounds for the variance of an estimator, based on observations selected with varying probabilities from a finite population, the elements of which are ranked with respect to the Y values. Accordingly, the usefulness of these bounds relates to the pre-enumeration analysis where one may well know the intended probabilities and joint probabilities corresponding to the sampling scheme but does not know the Y values. If, however, one can make a conservative guess at the largest Y value, one can use these bounds. Some examples are included to illustrate the theory. [ABSTRACT FROM AUTHOR]

It is desired to set confidence limits for the probability of successful operation at least until time x[sub 0] of a series system of k dissimilar components. The components follow exponential failure laws, and data are available on failure times of components of each type. Exact confidence limits for k=2 have previously been given by Lentner and Buehler. The present paper deals with a large-sample approximation to the exact solution for arbitrary k. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance component estimation for the unbalanced one-way random classification where the number (N[sub I]) of observations in the ith class is treated as a random variable not necessarily independent of the class effect (A[sub iota]). It is assumed that in general P(N[sub I] = 0) > 0. The conditional expectations (given the number of observations in each class) of all estimators of the between variance component (sigma[sup 2, sub alpha]) belonging to a certain class of estimators are derived. A general expression is found for the expected value of that estimator of sigma[sup 2, sub alpha] yielded by analysis of variance of class means. The limit of this expression (as the number of classes arrow right Infinity) is given; and it is shown that, if the bivariate distribution function of A[sub I], N[sub I] belongs to a certain class of distribution functions, then this limit is less than sigma[sup 2, sub a]. Numerical approximations to the expected values of two estimators of sigma[sup 2, sub a] are presented for one subclass of such distribution functions. [ABSTRACT FROM AUTHOR]

This paper carries on work in [2] and [4] on the asymptotic effects of small specification error in simultaneous equation estimation. The specification errors considered are inherent in the approximate nature of econometric models. It is shown that for any parameter and small enough specification error, the probability limits of two-stage least squares and limited-information maximum-likelihood, while different, lie on the same side of the true parameter value, the difference between them being of the second order of smalls relative to the effects of the specification error on either one of them. A method of consistently estimating the sign of the difference of the effects of specification error on the two estimators is given and a procedure suggested for choosing an estimator to reduce those effects. Two relatively simple examples are worked out. [ABSTRACT FROM AUTHOR]

In many linear regression problems, the values of the independent variable or variables may be subject to certain constraints. For example, the independent variables may necessarily be positive; as another example, the variables may not only all be positive but are powers of a single variable (e.g., polynomial regression on time). Previous writers considering the problem of obtaining confidence bands on a regression function for all values of the independent variable have not utilized such constraints; the usual basis for such bands has been the multiple comparison procedure of Scheffe which places no constraints at all upon the independent variables. Any procedure utilizing constraints will necessarily yield a uniform improvement over the method of Scheffe (assuming both methods are applicable) in the sense of yielding narrower bands for a given confidence probability. In the present paper a nontrivial lower bound is obtained for the confidence probability associated with a multiple comparison procedure appropriate to the case where it can be assumed that each independent variable must be of specified sign; this includes, as a subclass, polynomial regression on a non-negative independent variable. This result gives a basis for a multiple comparison procedure less conservative than that of Scheffe when both are applicable. Implementation of the procedure requires the percentage points of a heretofore untabulated distribution. Tables of percentage points of this distribution appropriate to linear combinations of two, three, or four parameters are presented. [ABSTRACT FROM AUTHOR]

The robustness properties of four estimators of location are studied with respect to eight distribution types. For each type, the probability density function is symmetric about the median and the range of variate is infinite. For the entire class of distributions, the estimator with the highest guaranteed efficiency is the mean of the middle fifty percent of the sample. This study supplements the paper by Crow and Siddiqui (1967). [ABSTRACT FROM AUTHOR]

While distribution-free procedures are often appropriate when testing statistical hypotheses, they may become complicated or involve loss of power when the data are grouped. For rank tests the ties caused by grouping are generally broken either by using a randomization procedure or averaging the tied ranks. In this paper the power loss due to equi-spaced grouping (in terms of Pitman asymptotic relative efficiency) is investigated for some commonly used tests, for each method of tie-breaking. The tests considered are Wilcoxon's and Mood's tests for the two-sample problem, Mann's test for randomness, and Pitman's independence test. It is shown how the power loss depends on the width of the grouping intervals and the distribution of the data, and some numerical studies are given. The results seem to indicate that the power loss is small even for a sizable group interval, and that it may be preferable to break ties by randomization than by averaging ranks. [ABSTRACT FROM AUTHOR]

This paper gives analytic expressions for the moments and recurrence relations for the first four raw moments of a doubly noncentral t-distribution with v degrees of freedom and noncentrality parameters δ and λ. Table I provides numerical values of these moments for v = 2 (I) 20, λ = 2 (2) 8 (4) 20 and any suitable σ. Two approximations to the t"-distribution, involving the moments, are considered. [ABSTRACT FROM AUTHOR]

Let Ω be an arbitrary set, a, a σ-field of subsets of Ω, and V(A), a finite, countably additive set function defined on a. Let A1, A2, ..., be a sequence of subsets of Ω belonging to a. Denote by Hk, k = 0, 1, 2, ..., the set of elements of Ω which belong exactly to k sets among A1, A2, .... In the theory of probability, in combinatorial analysis and in the theory of numbers there frequently arises the problem of finding V(Hk), k = 0, 1, 2 .... In this paper V(Hk) is found if V(Ω) and V(Ai1, Ai2 ... Ain, 1 ... i2 < i2 < ... < ir, are known. [ABSTRACT FROM AUTHOR]

The paper presents the results of an inquiry concerning the small sample relative efficiency of maximum likelihood and best unbiased estimators of reliability functions of one-unit systems. Three cases are considered: The Poisson, exponential, and normal (standard deviation known). Two kinds of relative efficiency functions are studied. The first kind consists of the common ratio of the Cramer-Rao lower bound of the variances of unbiased estimators, to the mean-square-error of the considered estimator. The second kind is a new type of a relative efficiency function, which is called 'the closeness relative efficiency function.' This function is defined as the ratio of the probabilities that the maximum likelihood and the beat unbiased estimators yield estimates in a prescribed neighborhood of the unknown reliability value. A substantial part of the study is devoted to the derivation of the required moments of the estimators. [ABSTRACT FROM AUTHOR]

Tins paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with "blocks" a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs. [ABSTRACT FROM AUTHOR]