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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]

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.

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]

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]

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]

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]

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]

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]

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]

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]

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 experiments to be discussed in this article was designed to investigate empirically the distribution of the sample version of the measure of association (G), &b.gamma;. Professors, William H. Kruskal and Leo A. Goodman have published an article giving interpretive motivation for the coefficient, and have developed the large sample theory for G. In the present research, sampling experiments were performed to test the adequacy of the large sample theory for 5X5 population cross classifications and samples of size ten, twenty-five and fifty. The numerical experiment described in this paper was based on a number of 25 cells multinomial populations, each considered as represented by the probabilities in a 5X5 population cross classification. Gamma is defined by Goodman and Kruskal in terms of probabilities. The 100 population cross classifications were chosen to be representative of practical situations in psychology. The method use to obtain them was to begin with 5X5 cross classification having complete association along the main diagonal and with a variety of marginal distributions.

In this paper the robustness of some of the existing tests for the exponential distribution when some outliers are present is reviewed. It is noted that the tests proposed by Epstein and Sobel [4] can be used without any major modification in the presence of outliers and a more suitable flexible test criterion is proposed for the exponential lower limit which is superior to Carlson's statistic [2] in all situations. The distribution of this statistic is derived and tables are constructed to facilitate the use of this statistic. Also some tests for outliers in the exponential case are proposed. [ABSTRACT FROM AUTHOR]

The use of Fourier or spectral methods for the seasonal adjustment of economic and other time series has recently been suggested; see for example Hannah [3], [4]. The case in which the seasonal pattern does not change appreciably from year to year has been covered in some detail but methods appropriate to an evolving seasonal pattern have been given only a little attention. The present paper discusses the nature of evolving seasonal patterns which may arise in economic series and examines a method due to Harman which may be used when the pattern is changing slowly over time. Although the suggested method is not readily mechanized for routine application on a large scale since some personal judgments are called for, it is found useful in detailed studies and might well be applied to any series for which traditional methods are not adequate; it is also applicable to the problem of prediction. [ABSTRACT FROM AUTHOR]

The determination of sample size to meet certain probability requirements is a problem which often faces an experimenter when obtaining a confidence interval or testing a hypothesis. This paper includes some new graphs useful in selecting sample size and gives a reference to a new textbook which includes a number of other similar type graphs. The method of construction is explained in detail and examples are included. [ABSTRACT FROM AUTHOR]

This article provides some corrections to the numerical results obtained in a recent paper by Duncan [2] and suggests a more efficient procedure for determining the optimum control parameters. [ABSTRACT FROM AUTHOR]