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Suppose that there are K regions in a country, with N[sub I] people and M, cities in the ith region. Let N[sub iota] be large, M[sub iota] random, given N[sub iota], and such that the distribution of M[sub I]N[sup -1, sub I], given N[sub I], converges to a limiting distribution F, with F(X) Is similar to Cx[sup gamma] as x arrow right 0, gamma > 0. Let L[sup (r)] be the size of the rth largest city in the country. If, given M[sub iota] and N[sub I], there is a Bose-Einstein allocation of the N[sub iota] people to the M[sub iota] cities in region I, independently for the various regions, then a plot of L[sup (r)] against r will be approximately proportional to r[sup -(1+alpha)], for 1 + alpha = gamma[sup -1]. [ABSTRACT FROM AUTHOR]

Two classification procedures are considered in estimating a proportion P in a dichotomous population. Given that classifiers may be subject to error, sufficient conditions are obtained on the probabilities of misclassification of the classifiers such that the bias of one procedure will always be smaller than the bias of the other procedure for typical values of P found in practice. The mean squared error of the former procedure is also shown to be superior for moderate sample sizes. [ABSTRACT FROM AUTHOR]

After providing some background for the need to consider estimates other than those resulting from normal theory, there is a brief review of some nonadaptive robust estimators. We introduce adaptive estimators using those of Tukey and McLaughlin, Jaeckel, Johns, Birnbaum and Mike, Takeuchi, Hajek, van Eeden, and Beran. Adaptive estimators based on preliminary testing and Stein-like procedures are then considered, and recommendations are made on how to select the amount of trimming. Various proposals for estimating regression coefficients are also considered. Adaptive distrubution-free tests look very promising for improving the power of nonparametric tests, and some of these techniques can be used effectively in data analysis. Asymmetric trimmed means, adapted to the particular sample, can easily be used with data and provide good descriptive statistics having an approximate error structure. Finally, it is conjectured that estimators based on "cliff-hangers" might be extremely effective if there are sharp changes in the height of the underlying density function. [ABSTRACT FROM AUTHOR]

Monte Carlo methods were used to study the performance of the largest-root test, two trace-type tests and three determinantal tests in the fixed-effects MANOVA model when certain assumptions are violated. Results indicated that for protection against nonnormality and heterogeneity of covariance matrices, the largest-root test should be avoided, while the Pillai-Bartlett trace test may be recommended as the most robust of the MANOVA tests, with adequate power to detect true differences in a variety of situations. [ABSTRACT FROM AUTHOR]

Several one-pass and two-pass algorithms for the computation of sample means and variances are compared by their performance on sets of randomly generated data and systematically generated data with random noise. The relation between the performance of each algorithm and the coefficient of variation of the population from which random data sets are generated is explored. [ABSTRACT FROM AUTHOR]

A technique is presented for testing the hypothesis that a hierarchical sequence of partitions constructed by the single-link or complete-link clustering method could have been obtained because of "noise." Two rank orderings of the object pairs are compared. One Of the orderings is obtained from the initial proximity values; the second is derived from the levels at which an object pair first appears within a single subset within the hierarchy. The hypothesis that the given set of proximity values have been assigned randomly is tested by referring the Good- man-Kruskal rank correlation γ statistic to an approximate permutation distribution. [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]

Three statistics for testing additivity and making detailed inferences on interactions in a two-way setup are considered. The first method uses the well-known sum of squares for testing and yields Scheffe bounds for all contrasts; the second introduces a maximum characteristic root statistic and Roy bounds, but is restricted to product-type contrasts; the last is limited to tetrad differences and its statistic is the maximum of all such differences. Bounds on percentage points for this maximum are provided in the appendix. Uses of the three methods in inferring various types of interaction contrasts are discussed and illustrated. [ABSTRACT FROM AUTHOR]

In the nonparametric comparison of two treatments using paired data, one frequently obtains zeros when computing the absolute difference for each of the N pairs. While Wilcoxon recommends omitting the zeros, Pratt argues that all N differences should be ranked and that only the ranks associated with the nonzero differences (n) be used in the test. Tables of critical values of the Pratt signed rank statistic are presented for n = 5(1) 50 nonzero differences and Z = 0(1) 20 difference-scores of zero at two-tail probability levels of .10, .05, .02, and .01. [ABSTRACT FROM AUTHOR]

Suppose that Y[sub 0],..., Y[sub q] are q + 1 criterion variates and that X is a p-dimensional column vector of predictor variates. A principal result of this article is the analytical solution of the problem of determining a p X I vector b to maximize the correlation rho(Y[sub 0], b'X) subject to the constraints rho(Y[sub i], b'X) = w[sub i], i = 1,..., q, where w[sub 1],..., w[sub q] are specified constants. Some frequently encountered selection problems can be solved by applying this result. [ABSTRACT FROM AUTHOR]

In aerial census data "visibility bias" is present because of the failure to observe some animals. A model is presented for quadrat sampling of randomly occurring groups whose size follows a single parameter power series distribution when there is a probability q > 0 of missing single animals. Maximum likelihood estimates of group density, average group size, and the visibility bias parameter, q, are presented. Examples with moose census data are worked out. [ABSTRACT FROM AUTHOR]

of a continuous autoregressive moving average process at an equispaced sampling interval results in a discrete autoregressive moving average process. The relationship between the continuous and the discrete parameters yields a simple method of maximum likelihood estimation of the continuous parameters from a discretely sampled data. A technique is described for modeling of continuous processes from discrete observations and is illustrated with analysis of the yearly Wolfer's sunspot numbers data. [ABSTRACT FROM AUTHOR]

It is shown that, under general conditions, rational random behavior (appropriately defined) is equivalent to large-sample maximum-likelihood estimation of the theoretically optimal decision. Some supporting evidence, based on Dutch and American consumption data, is presented. The approach can also be used as a statistical basis for the delegation of authority. [ABSTRACT FROM AUTHOR]

A procedure which utilizes an initial estimate to estimate the coefficients in a linear mode is introduced. A linear model containing replications with the same input matrix is considered and a class of twostage estimators is defined. Optimal estimators in the class are determined, and it is shown that the proposed estimators are better than the corresponding single-sample least squares estimator if the initial estimate is accurate, and are as good (asymptotically) as the corresponding single-sample least squares estimator if the initial estimate is inaccurate. [ABSTRACT FROM AUTHOR]

We show that the power of the noncentral F-test increases with the number of denominator d.f. and decreases with the numerator d.f. Hence, increasing the dimension of the observed vectors may decrease the power of Hotelling's T[sup 2]-test for means, unless the additional noncentrality provided by the extra variates is sufficiently large. A function determining the exact increase in noncentrality required for increased power is studied and tabulated We present procedures designed to test, on the basis of a preliminary sample, whether the increase in noncentrality is sufficiently large to justify inclusion of the extra variates. [ABSTRACT FROM AUTHOR]

In this article, we consider a system of two "seemingly unrelated regression'' equations (y[sub i] = X[sub i]beta[sub i] + u[sub i], i = 1, 2) with X is equivalent to (X[sub 2] L[sub 1]), and examine some finite sample properties of beta-estimators based on the unrestricted estimate $ of SIGMA = (sigma[sub 2 j]). The estimator beta[sub 2] of beta[sub 2] is shown to be identical with the direct OLS estimator b[sub 2] obtained from Equation 2. The estimator beta[sub 1] of beta[sub 1] is shown to be more efficient than b[sub 1], the direct OLS estimator of beta[sub 1] obtained from Equation 1, for moderate departures of rho = sigma[sub 12] square root of sigma[sub 11]sigma[sub 22]from zero; the efficiency, moreover, is shown to increase rapidly with the sample size for any given rho. [ABSTRACT FROM AUTHOR]

A group test is a single test on a group of items which indicates either that the group is free of defectives or that there is at least one defective present. Sobel and Groll [2] have given a test procedure which is based on a set of recursive equations, called the R[sub I] procedure, for finding all the defectives in a finite population. This procedure is extended to the case where there is a known positive probability of the test yielding an erroneous result when the test group contains one or more defectives. [ABSTRACT FROM AUTHOR]

In this article we derive the exact finite sample distribution of a statistic utilized by Theil [10] to test whether sample and prior information are in conflict with each other. A weighted estimator for the slope coefficient in a simple regression equation with one independent variable is obtained by using weights which are determined by the observed value of Theil's test statistic. This weighted estimator is particularly useful when we are uncertain about the compatibility of prior information with sample information. [ABSTRACT FROM AUTHOR]

Kumar and Sobel studied the problem of finding a single defective in an infinite binomial population by group testing. Let E(T) denote the minimum expected number of group tests needed to find one defective unit if the units come from an infinite population. If a group of m units from this population is tested and found defective, then let F(m) denote the minimum expected number of tests to find one defective unit in this group. This article proves two conjectures made by Kumar and Sobel concerning E(T) and F(m) and gives a closed-form solution for F(m). [ABSTRACT FROM AUTHOR]

The problem considered is how to compare time-to-response data for different groups when the group membership of an individual can be arbitrarily varied during a study. Modified life tables can be constructed which reflect such changes in an individual's status, and associated measures of relative risk and statistical significance calculated. This is illustrated with survival data for heart-transplant patients, for which a patient can transfer from the nontransplanted to the transplanted group. Alternative procedures are given in which distinctive groups are defined for each transplant day. [ABSTRACT FROM AUTHOR]

This article proposes a method of discounting evidence when linear regression models are constructed after the data have been partially, analyzed. The solution parallels a formal decision theoretic analysis of a presimplification problem in which models or model spaces are simplified before observation in order to avoid observation or processing costs. Post-data model construction is interpreted as the data-dependent decision that this pre-simplification is undesirable. The discounting implications of this formal analysis may then be used to police data analyses with less formal post-data model construction. [ABSTRACT FROM AUTHOR]

This article applies the counting rule strategy to the problem of estimating the proportion of population elements with a specified characteristic based on a sample of enumeration units. The distribution of population elements among the enumeration units is a function of the counting rule which specifies the conditions for linking population elements to enumeration units. The counting rule strategy involves the selection of the counting rule which minimizes the variance of the estimate for fixed cost. The strategy is illustrated by a problem which involves estimating the proportion of incorrect statistical statements in the text of a report based on a sample of lines of text. [ABSTRACT FROM AUTHOR]

Hogg [2, 3, 4], Hollander [5], and Randles and Hogg [10] have obtained interesting results about zero correlation and certain symmetry properties for classes of statistics. One of the purposes of this article is to collect some of their results and extend them where possible. Several examples are presented demonstrating the broad applicability of the extended results. [ABSTRACT FROM AUTHOR]

Consider a problem in which a certain statistic X has a specified distribution function F(x) if a given hypothesis H is true, and suppose that the hypothesis H is evaluated by calculating the tail area 1 -- F(x) corresponding to the observed value x of the statistic X. Examples are given in which this tail area is equal to the posterior probability that H is true and in which it is equal to the likelihood ratio comparing H to a certain class K of alternatives. The purpose of these examples is to render the traditional statistical practice of calculating tail areas consonant with the principles of Bayesian statistics. [ABSTRACT FROM AUTHOR]

This is an application of a least-squares prediction approach to finite population sampling theory. One way in which this approach differs from the conventional one is its focus on characteristics of particular samples rather than on plans for choosing samples. Here we study samples in which many superpopulation models lead to the same optimal (BLU) estimator. Random sampling is considered in the light of these results. [ABSTRACT FROM AUTHOR]

This article reviews some of the history of robust estimation, emphasizing the period 1885-1920 and the work of Newcomb, Edgeworth, Sheppard, and Daniell. Particular attention is paid to lines of development which have excited recent interest, including linear functions of order statistics and mixtures of normal densities as models for heavy-tailed populations. [ABSTRACT FROM AUTHOR]

Given a set of grouped survival data, least squares estimates are proposed for the parameters of four survival distributions that can be fit: exponential, linear hazard, Gompertz and Weibull. Sample estimates of the hazard function are utilized in the least squares procedures and a method is given for selecting a distribution for further investigation based on the likelihood under the four survival models. A Monte Carlo study demonstrated that the least squares estimates are nearly as efficient as maximum likelihood when the sample size is 50 or more. The methods are applied to survival data for 112 patients with plasma cell myeloma. [ABSTRACT FROM AUTHOR]

How can you use a product-moment correlation coefficient when you do not have interval scale data? This is particularly addressed to uses of such statistics with personality, value, or other 'soft' variables, and to variables that are sometimes considered to be composed of discrete categories. The implicit contrast usually is to a nonparametric or sidtribution free statistic like Maurice G. Kendall's Tau as more appropriate with such data. Since part of the confusion that surrounds the question of the appropriateness of use of various statistics occurs about the type of measurement involved, we may focus on the presentation of Herbert M. Blalock on this topic.

This article considers the familiar linear model y = X beta + e in which the residual vector e has elements e, with expectation 0 but different variances sigma[sup 2, sub I]. The elements of beta and the sigma[sup 2, sub I] are estimated by maximum likelihood under the assumption of a lower bound for the sigma[sup 2, sub I] of the form 0 < sigma[sup 2, sub I] is greater than or equal to sigma[sup 2, sub 1] so that the likelihood is finite in the restricted parameter space. A second problem is considered in which the y vector splits into k sub-vector y[sub 1] such that all elements of y[sub j] have equal variances. Various asymptotic optimality properties and small sample unbiassedness for beta are proved. [ABSTRACT FROM AUTHOR]

Stein's estimator for k normal means is known to dominate the MLE if k is greater than or equal to 3. In this article we ask if Stein's estimator is any good in its own right. Our answer is yes: the positive part version of Stein's estimator is one member of a class of "good" rules that have Bayesian properties and also dominate the MLE. Other members of this class are also useful in various situations. Our approach is by means of empirical Bayes ideas. In the later sections we discuss rules for more complicated estimation problems, and conclude with results from empirical linear Bayes rules in non-normal cases. [ABSTRACT FROM AUTHOR]

This study is concerned with deriving the properties of the preliminary test estimator for the general linear normal regression model, ascertaining the characteristics of the risk functions over the parameter space, and determining the conditions necessary for the risk of this estimator to exceed or be less than the conventional one under squared error loss. A test procedure and the problem of choosing an optimal level of significance for the test are discussed. Some theorems and lemmas used in evaluating the risk and some properties of functions of the non-central F distribution are developed in the appendices. [ABSTRACT FROM AUTHOR]

The statistics of several alternative procedures, proposed to test for independence in contingency tables derived from stratified samples, all have the same asymptotic chi-square distribution, under the null hypothesis. By considering a modified sampling procedure, it is shown here that they also have the same asymptotic distribution under any alternative hypothesis--the noncentral chi square, with the noncentrality parameter of Wald's test for the case of independently and identically distributed variables. Simulation, used to compare the finite-sample powers of the tests, shows small differences between the various methods and close approximation to the asymptotic results. [ABSTRACT FROM AUTHOR]

Dempster's method of upper and lower probabilities represents an attempt to obtain posterior probability statements about parameters without the necessity to incorporate prior information. We apply Dempster's method to the case of multinomial data for which there is some natural ordering of the categories. Inferences are derived for the sum of an arbitrary set of categories, the ratio of two such sums, and linear combinations of the category probabilities. Numerical examples are given, and some effort is made to interprete Dempster's approach and relate it to the standard approaches to statistical inference. [ABSTRACT FROM AUTHOR]

This article discusses the need for and advantages of test data expressible in mathematical form. The need to test computer programs for statistical calculations frequently arises and, depending upon circumstances, it may be necessary to test the program logic, to examine the computational error of the algorithm, or to assess comprehension of instructions for using the program. The adequacy with which the programs may be tested depends to a large extent on the availability of suitable test data. For balanced complete designs which are nested, crossed, or both nested and crossed, test data can be conveniently expressed in terms of orthonormal polynomials. It is especially important that test data be available for the most widely used procedures in statistics. Examples of test data are given for least squares and ANOVA in the article and suggestions are made for constructing test data for some other simple statistical procedures. Programs for complex statistical procedures are usually much more flexible to use and are easier to test if they are written in modular form. The kind of test data authors are considering here is easier to construct for individual modules than for a long string of modules, where it is unlikely that one could obtain simple expressions for the outputs of the final module of such a string given an expression for some particular input to the first module.

To judge from articles and research notes in seven political science journals published over a two-and-one-half year period ending in the summer of 1970, political science has markedly increased in its reliance on quantitative data and statistical and mathematical techniques. Thus the percentage of quantitative articles in the American Political Science Review has risen from 12 percent to 65 percent, in a 25-year period, while use of "more powerful" techniques has risen from less than 1 percent to 53 percent. Statisticians' attention is directed to the difficulties associated with political science data. [ABSTRACT FROM AUTHOR]

Systematic procedures for constructing confidence bounds and point estimates based on rank statistics are given for the two sample location parameter, two sample scale parameter and one sample location parameter problems. [ABSTRACT FROM AUTHOR]

The article shows that if the original variable follows a pth order autoregressive system then the non-overlapping moving sum follows a pth order autoregression with at most a pth order moving-average of an independent sequence regardless of the length of the summation. From such an aggregate model we derive the optimal predictor of the aggregate variable and show that it performs remarkably well compared to the optimal disaggregate predictor. The article contains both theoretical and numerical analysis. [ABSTRACT FROM AUTHOR]

This article is concerned with choosing a "best" one from a class of available random alternatives, using this simple probability model: corresponding to the ith alternative is a real-valued random variable L4, and our choice is to be based solely on the joint distribution of the L1's, which is known. This simple model can be a reasonable one in the following real situations, which will be used as examples: speed contests, elections, taste preferences, choice from among possible strategies in a game or wagers in gambling, choice from among possible statistical decision procedures. This simple model will not suffice when preferences among realized alternatives cannot be based simply on one number for each, as happens in statistics when there are several kinds of loss that cannot be freely traded one for another. Nor will this simple model do when unknown distributions are involved, as is usual in statistics except in Bayesian statistics. The randomization device used to generate X and V is not explicitly described in utility theory, and the concealed assumption exists that preference between x and y can be based solely on the separate marginal distributions of X and V.

The asymptotic sampling theory discussed in our 1963 article [3] for measures of association presented in earlier articles [1, 2] turns on the derivation of asymptotic variances that may be complex and tedious in specific cases. In the present article, we simplify and unify these derivations by exploiting the expression of measures of association as ratios. Comments on the use of asymptotic variances, and on a trap in their calculation, are also given. [ABSTRACT FROM AUTHOR]

When both variables in a normal regression model are masked by measurement errors of known variances, it is possible to obtain inferences about the parameters relating to the regression of the true variables. For two-sample problems, this article proposes tests of hypotheses of equality of (1) the regression coefficients and of (2) the residual variances, assuming o joint normal distribution on the predictor and predicted variables. [ABSTRACT FROM AUTHOR]

This study concerns the relative efficiency of the least-squares estimator of the mean of a covariance stationary sequence that has a constant mean and an autoregressive representation of order p. Our results augment those in [2]. The results in Section 1 include for any positive integer p the variance of the LSE, the BLUE in terms of autoregressive coefficients, the variance of the BLUE and the range of applications of Ylvisaker's theorem [7] regarding a lower bound on relative efficiency. For second-order schemes it also includes bounds on parameter values for relative efficiency is greater than or equal to 0.80. Graphical results for the first-order scheme in Section 2 add to the detailed study made in [2] and imply that the relative desirability of the LSE should be based on variance as well as relative efficiency considerations. [ABSTRACT FROM AUTHOR]

Half sample replication has been used as a method for estimating variances in stratified sampling whereby two primary selections are drawn from each stratum. This article considers some procedures of improving the precision of the variance estimator computed from a partially balanced set of half samples. Both the theoretical and empirical findings suggest that the SAOA procedure is an effective method to employ, in particular with a "2-order partially balanced design." [ABSTRACT FROM AUTHOR]

The effects of using survival ratios of different degrees of accuracy to estimate the net migration of the male Negro populations of 25 southern counties and six southern states during the three intercensal periods from 1930-1960 are examined. First, the differences within units (states and counties) produced by the four different sets of survival ratios are examined; and second, the differences between units are considered. In both cases differences are largest during the 1930-1940 decade and decrease at each successive intercensal period. [ABSTRACT FROM AUTHOR]

Let Y[sub ij] (i = 1, ... , l; j = 1, ... , J) be independent Poisson random variables, with expectations lambda[sub ij] = alpha[sub i], beta[sub j], where SIGMA alpha[sub i] = l. Such a model is called a multiplicative Poisson; its factor parameters alpha[sub i], are called severity factors, and beta[sub j] are called intensity factors. The article concentrates on the problem of estimating the severity factors alpha[sub i]. Two types of estimators of alpha[sub i] are derived, Bayes and least-squares-maximum-likelihood. The expectations and mean square errors of these estimators are given and their relative efficiency is tabulated. The intensity factors beta[sub j] are estimated from T[sub j] = SIGMA[sup I, sub i=1] Y[sub ij], which are independent and have Poisson distributions with expectations theta[sub j] = l beta[sub j], according to the common procedures. [ABSTRACT FROM AUTHOR]

Under an appropriate loss function, interval estimation may be regarded as a Bayesian decision-making procedure in which the objective is to find an interval that minimizes expected loss. For various loss functions, the behavior of the optimal interval is investigated, a comparison is made with the usual non-decision-theoretic interval estimates, and applications and examples are discussed. [ABSTRACT FROM AUTHOR]

Employing the small-disturbance approach, I study the bias and mean squared error of a class of estimators of the variance of the disturbances in a simultaneous linear stochastic equation model. The estimator considered is the residual sum of squares after estimation of the structural parameters by a k-class estimator, divided by an arbitrary scalar. The scalar which minimizes the mean squared error is (T-n+2), regardless of which k-class estimator is used, where T is the number of observations and n is the number of structural parameters estimated. Using the scalar (T-n+2), the member of the k-class that minimizes the mean squared error is classical least squares. [ABSTRACT FROM AUTHOR]

This article is concerned with the exact finite-sample distribution of the limited-information maximum likelihood estimator when the structural equation being estimated contains two endogenous variables and is identifiable in a complete system of linear stochastic equations. The density function derived, which is represented as a doubly infinite series of a complicated form, reveals the important fact that For arbitrary values of the parameters in the model, the LIML estimator does not possess moments of order greater than or equal to one. [ABSTRACT FROM AUTHOR]

Sobel and Weiss [2] have given an inverse sampling stopping rule for selecting the better of two binomial populations. Sampling is done by the play-the-winner method. By including the number of failures in the Sobel-Weiss stopping rule a truncated procedure is obtained with good general properties. In particular, excessive sample sizes are avoided when the population probabilities are small. [ABSTRACT FROM AUTHOR]

In the construction of econometric models, the need frequently arises to test the significance of coefficients whose distribution is a ratio of two normal random variables for which no statistical tables exist. In this article the distribution of such a ratio, emanating from a partial adjustment model, is empirically approximated by utilizing Monte Carlo experiments with various sample sizes. The results indicate that the standardized distribution of the ratio h positively skewed and does not closely approximate the z-distribution. On the basis of the empirical distribution obtained, critical values for various levels of significances and sample sizes are reported. [ABSTRACT FROM AUTHOR]

We discuss compromises between Stein's estimator and the MLE which limit the risk to individual components of the estimation problem while sacrificing only a small fraction of the savings in total squared error loss given by Stein's rule. The compromise estimators "limit translation" away from the MLE. The calculations are pursued in an empirical Bayesian manner by considering their performance against an entire family of prior distributions on the unknown parameters. [ABSTRACT FROM AUTHOR]