We consider the general linear model with independent symmetric errors. In this context we propose and examine the large sample behavior of some estimates of the regression parameters. For the location model these statistics are linear combinations of order statistics. In general they depend on a preliminary estimate and the ordered residuals based on it. The asymptotic efficiency of these procedures is independent of the design matrix. Specifically analogues of the median and trimmed and Winsorized means are proposed.

A general model is proposed for analysis of frequency tables. This model includes conventional log-linear models for complete and incomplete factorial tables and logit models for quantal response analysis. By use of coordinate-free methods of linear algebra and differential calculus, complete minimal sufficient statistics and likelihood equations for the maximum likelihood estimate are derived. The maximum likelihood estimate is shown to be unique if it exists, and necessary and sufficient conditions are given for its existence.

The generalized likelihood ratio is used to define a stopping rule for rejecting the null hypothesis $\theta = \theta_0$ in favor of $\theta > \theta_0$. Subject to a bound $\alpha$ on the probability of ever stopping in case $\theta = \theta_0$, the expected sample sizes for $\theta > \theta_0$ are minimized within a multiple of $\log \log \alpha^{-1}$, the multiple depending on $\theta$. An heuristic bound on the error probability of a likelihood ratio procedure is derived and verified in the case of a normal mean by consideration of a Wiener process. Useful lower bounds on the small-sample efficiency in the normal case are thereby obtained.

From a finite population units are drawn with varying probabilities with replacement. There is a certain cost for observing a unit. In this paper samples are obtained partly by drawing a fixed number of times, and partly by drawing and observing units until the cost reaches a specified level. Let $X_k$ be the number of times the $k$th unit has been drawn in either case. Consider for a given function $g(\bullet)$ the random variable $Z = \sum_k g(X_k, k)$. Under general conditions it is proved that $Z$ is asymptotically normally distributed (actually a multidimensional generalization is considered). By appropriate choices of $g(\bullet)$ asymptotic distributions are obtained in successive sampling with varying probabilities without replacement and for the mean of the distinct units in a simple random sample with replacement. It is also investigated how heterogeneous catchability and effects of marking affect the "Petersen" estimator in capture-recapture theory.

Among all sequential tests with prescribed error probabilities of the null hypothesis $H_0: \theta = -\theta_1$ versus the simple alternative $H_1: \theta = \theta_1$, where $\theta$ is the unknown mean of a normal population, we want to find the test which minimizes the maximum expected sample size. In this paper, we formulate the problem as an optimal stopping problem and find an optimal stopping rule. The analogous problem in continuous time is also studied, where we want to test whether the drift coefficient of a Wiener process is $-\theta_1$ or $\theta_1$. By reducing the corresponding optimal stopping problem to a free boundary problem, we obtain upper and lower bounds as well as the asymptotic behavior of the stopping boundaries.

Often, the probability of the simultaneous occurrence of dependent events can be well approximated by assuming them to be independent. Here, we discuss bounds on the error in using this procedure and conditions when under (or over) estimates occur. An inequality involving expectations of conditionally independent random variables is proven. Applications treat extreme values of exchangeable random variables and error probabilities for simultaneous inference. Bivariate dependence concepts treated by Lehmann are generalized to the multivariate case in such a way that relations valid for the bivariate case continue to hold.

This paper describes a general method for deriving optimal procedures for problems where the covariance matrices are patterned under both null and alternative hypotheses. The pattern considered in this paper was first suggested by Olkin (1970) and is a generalization of the intraclass correlation model of Wilks (1946) and arises in the study of interchangeable random variables. We prove a theorem showing how we can transform most such problems to products of problems where the covariance matrices are unpatterned. This theorem is applied to two problems, the multivariate analysis of variance problem and the multivariate classification problem where in both cases the covariance matrix is assumed patterned. We use theorems about products to derive optimal procedures for these problems. We then look at Olkin's pattern for the mean vector, and show that most problems where both the mean vector and covariance matrix are patterned can be transformed to a product of problems, one of which is trivial. The same two examples are studied where now both mean vectors and covariance matrices are assumed patterned. We also consider the problem of testing that the mean vector is patterned when we know the covariance matrix is.

Some new test criteria are proposed for testing various hypotheses concerning covariance matrices. Asymptotic expansions of their null distributions are derived in terms of the $\chi^2$-distribution.

Let $X = (X_1, \cdots, X_n)$ where the $X_i: p \times 1$ are independent random vectors, and let $A: n \times n$ be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix $XAX'$ be positive definite w.p.1. The results are applied to cases where $A$ has a particular form or $X_1, \cdots, X_n$ are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix $\sigma(X_i - \bar{X})(X_i - \bar{X})'$ is positive definite w.p. 1 $\operatorname{iff} P\lbrack X_1 \in F\rbrack = 0$ for every proper flat $F \subset R^p$.

Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for testing the equality of a covariance matrix, equality of a mean vector and a covariance matrix, independence between two sets of variates, equality of two covariance matrices, in multivariate normal distributions are derived under the sequence of alternative hypotheses converging to the null hypothesis when the sample size tends to infinity. Numerical accuracies of the asymptotic formulas are also examined.

This paper studies the relationship between games of search on a compact metric space $X$ and the absolute epsilon entropy $I(X)$ of $X$. The main result is that $I(X) = -\log \nu_L^\ast, \nu_L^\ast$ being the lower value of a game on $X$ we call "restricted hide and seek."

This note gives an extension of the Hajek-Renyi inequality without the assumptions of independence and of the moment conditions. It is a generalization of results of Kounias and Weng [3].

In this paper, we have proved a fundamental property of the characteristic function for the random variable $(\partial/\partial\theta) \log f(x \mid \theta)$. Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators $\{\theta_n^\ast\}$ are asymptotically efficient in Bahadur's sense $\lim_{\varepsilon \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n\varepsilon^2} \log P\theta\{|\theta_n^\ast - \theta| \geqq \varepsilon\} = -\frac{I(\theta)}{2}.$ Our proof also gives a simple and direct method to verify Bahadur's [2] result.

Let $X = (X_1, \cdots, X_k)$ be a random vector whose distribution depends on a parameter vector $\theta = (\theta_1, \cdot, \theta_k)$. A standard procedure $\phi^\ast$ is considered for selecting a set of $m < k$ coordinate values corresponding to the $m$ largest components of $\theta$. $\phi^\ast$ is given as follows: Select the $m$ coordinates corresponding to the $m$ largest components of $x$, the observed value of $X$. Break ties, if any, with randomization. Some optimal properties of $\phi^\ast$ are known, given that the loss function and the distribution of $X$ have certain invariance and monotonicity properties. It is shown in this paper that $\phi^\ast$ is a Bayes decision rule if $X$ is "stochastically increasing" in $\theta$.

Let $X_1, \cdots, X_n$ be i.i.d. $N(i, 1), i = 0, \pm 1, \pm 2,\cdots$. Hammersley [2] proposed $\lbrack\bar{X}_n\rbrack$, the nearest integer to the sample mean, as an estimator of $i$. It is proved that $d$ is minimax and admissible relative to zero-one loss. However, it is shown that relative to squared error loss, the estimator is neither admissible nor minimax.

This paper shows that a quadratic form in a multivariate sample has a certain rank and its nonzero eigenvalues are distinct with probability one under the assumption that the matrix defining the quadratic form satisfies a certain rank condition and that the underlying distribution of the sample is absolutely continuous with respect to Lebesgue measure.

Conditions are derived which are commonly met in applications and which are sufficient for both one- and two-sided Scheffe bounds to have no greater average width than any other-shaped confidence bounds which have the same confidence coefficient.

Various selection problems based on sample means have been considered by Bechhofer (Ann. Math. Statist. (1954), 16-39). Parallel selection procedures based on one-sample Hodges-Lehmann estimators (Ann. Math. Statist. (1963), 598-611) are proposed in this paper. These procedures possess more desirable finite sample properties and equivalent asymptotic properties as compared to rival procedures considered by Lehmann (Math. Ann. (1963), 268-275) and Puri and Puri (Ann. Math. Statist. (1969), 619-632).

A homogeneous random process on the sphere $\{X(P): P \in S_2\}$ is a process whose mean is zero and whose covariance function depends only on the angular distance $\theta$ between the two points, i.e. $E\lbrack X(P)\rbrack \equiv 0$ and $E\lbrack X(P)X(Q)\rbrack = R(\theta)$. Given $T$ independent realizations of a Gaussian homogeneous process $X(P)$, we first derive the exact distribution of the spectral estimates introduced by Jones (1963 b). Further, an estimate $R^{(T)}(\theta)$ of the covariance function $R(\theta)$ is proposed. Exact expressions for its first- and second-order moments are derived and it is shown that the sequence of processes $\{T^{\frac{1}{2}}\lbrack R^{(T)}(\theta) - R(\theta)\rbrack\}^\infty_{T=1}$ converges weakly in $C\lbrack 0, \pi\rbrack$ to a given Gaussian process.

Rees [2] introduced the concept and name of "neighbor" designs. The problem can be described as that of arranging $v$ kinds of objects on $b$ plates each containing $k$ objects in a loop such that every object on a plate has two neighbors. The requirements are that each object appears $r$ times (but not necessarily on $r$ different plates) and is a neighbor of every other object exactly $\lambda$ times. This paper constructs neighbor designs with parameters as follows: $(i) k > 2:\quad v = 2k + 1, \lambda = 1$ $(ii) k \equiv 0 (\mod 2) > 2:\quad v = 2^i k + 1, i = 1,2,\cdots, \lambda = 1$ $(iii) k \equiv 0 (\mod 4):\quad v = 2mk + 1, m = 1,2,\cdots, \lambda = 1.$

The value of a zero-sum two-person game with infinite number of stages can be defined either directly or as the limit of the values $v_n$ of the truncated games with $n$ stages. It is shown that these two concepts are not equivalent. There are games in which $\lim v_n$ exists but which do not have values as infinite stage games.