A model for random $n \times k$ matrices $X$ is considered. The elements $X_{ij}$ are assumed to be independent and Poisson-distributed random variables with means $\alpha_i\exp(\gamma t'_{ij})$, where $t_{ij}$ are known $m$-dimensional vectors and $(\alpha, \gamma)$ is an unknown parameter. A goodness-of-fit test is proposed and an approximation is derived when the number $n$ of rows is large.

A maximum likelihood fit of a logistic regression model (and other similar models) is extremely sensitive to outlying responses and extreme points in the design space. We develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit. The elements of the fitting process which constitute the usual output (parameter estimates, standard errors, residuals, etc.) will be used for this purpose. With a properly designed computing package for fitting the usual maximum-likelihood model, the diagnostics are essentially "free for the asking." In particular, good data analysis for logistic regression models need not be expensive or time-consuming.

Asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors. These asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest. Applications of the general results to principal components analysis and canonical variate analysis are given.

In this paper designs are found which are optimum for various models that include some autocorrelation in the covariance structure $V$. First it is noted that the ordinary least squares estimator is quite robust against small perturbations in $V$ from the uncorrelated case $V_0 = \sigma^2_I$. This "local" argument justifies our use of such estimators and restriction to the class of designs $\mathscr{X}^\ast$ (balanced incomplete block or Latin squares) optimum under $V_0$. Within $\mathscr{X}^\ast$ we search for designs for which the least squares estimator minimizes appropriate functionals of the dispersion matrix under various correlation models $V$. In particular, we consider "nearest neighbor" correlation models in detail. The solutions lead to interesting combinatorial conditions somewhat similar to those encountered in "repeated measurement" designs. Typically, however, the latter need not be BIBD's and require twice as many blocks. For Latin squares, and hypercubes, the conditions are less restrictive than those giving "completeness."

The maximum likelihood estimate is shown to exist and to be unique if a twice continuously differentiable likelihood function is constant on the boundary of the parameter space and if the Hessian matrix is negative definite whenever the gradient vector vanishes. The condition of constancy on the boundary cannot be completely removed, cf. Tarone and Gruenhage (1975). The theory is illustrated with several examples.

According to an invariance principle, for some models having a certain group structure, there is a uniquely defined prior representing ignorance, which is called the inner prior. It is shown that the corresponding posterior probability of a likelihood region has a simple frequency interpretation as a mean conditional confidence level. The central multivariate normal model is considered as an example.

The problem of selecting a random subset of good populations out of $k$ populations is considered. The populations $\Pi_1, \cdots, \Pi_k$ are characterized by the location parameters $\theta_1, \cdots, \theta_k$ and $\Pi_i$ is said to be a good population if $\theta_i > \max_{1 \leq j\leq k}\theta_j - \Delta$, and a bad population if $\theta_i \leq \max_{1 \leq j \leq k} \theta_j - \Delta$, where $\Delta$ is a specified positive constant. A theory for a special class of procedures, called Schur procedures, is developed, and applied to certain minimax problems. Subject to controlling the minimum expected number of good populations selected or the probability that the best population is in the selected subset, procedures are derived which minimize the expected number of bad populations selected or some similar criterion. For normal populations it is known that the classical "maximum-type" procedures has certain minimax properties. In this paper, two other procedures are shown to have several minimax properties. One is the "average-type" procedure. The other procedure has not previously been considered as a serious contender.

Using Roy's union-intersection principle, a unified treatment is developed for the construction of multivariate tests. These include Wilks' determinantal criteria, Hotelling-Lawley trace criterion, and Roy's largest characteristic root criterion. The key lies in the extension of an index set from vectors to matrices plus the use of elementary symmetric functions of characteristic roots to test component hypotheses.

The posterior distribution of a Dirichlet process from $N$ quantal responses at $r$ dosage levels has been recognized by Antoniak as a mixture of Dirichlet processes. The purpose of this paper is to develop a systematic procedure for computing finite-dimensional distributions of such mixtures which can be equivalently expressed as multivariate beta distributions with random parameter vectors. It is shown that the sequence of random parameter vectors of the updated beta posteriors from observations at increasing dosage levels evolves in a manner which is described by $r$ separate Markov chains. This description is then used to derive the asymptotic posterior distribution. The weak limits of the relevant Markov chains are shown to be solutions of certain stochastic differential equations and the random parameter vector of the posterior beta distribution is shown to be asymptotically normal, the mean vector and covariance matrix of which are given by recursion formulas.

The following decision problem is studied. The statistician observes a random $n$-vector $y$ normally distributed with mean $\beta$ and identity covariance matrix. He takes action $\delta\in\mathbb{R}^n$ and suffers the loss $L(\beta, \delta) = (\beta'\delta - 1)^2.$ It is shown that this is equivalent to the linear control problem and closely related to the calibration problem. Among the invariant estimators, it is shown that the formal Bayes rules together with some of their limits include all admissible invariant rules. Other results on admissibility and inadmissibility of some commonly used estimators for the problem are obtained.

The distribution function of a random variable of the form $\sum^n_{i = 1} a_i Y_1 Y_2 \cdots Y_i$ where $a_i > 0$ and $0 \leq Y_i \leq 1$ is considered. A geometric argument is used to obtain the distribution function as a repeated integral. The result is used first to obtain the distribution function of a linear combination of variables defined over the simplex $X_i \geq 0, \sum^n_{i = 1} X_i \leq 1$. As a second application the distribution of certain quadratic forms over the simplex is obtained. This result yields as a special case the distribution of the internally studentized extreme deviate; the cases of normal and exponential samples are considered in detail and the required distributions obtained.

Consider the problem of sequentially testing a null hypothesis vs an alternative hypothesis when the risk function is a linear combination of probability of error in the terminal decision and expected sample size (i.e., constant cost per observation.) Assume that the parameter space is the union of null and alternative, the parameter space is convex, the intersection of null and alternative is empty, and the common boundary of the closures of null and alternative is nonempty and compact. Assume further that observations are drawn from a $p$-dimensional exponential family with an open $p$-dimensional parameter space. Sufficient conditions for Bayes tests to have bounded stopping times are given.

Let $X$ be a random variable which takes on only finitely many values $x \in \chi$ with a finite family of possible distributions indexed by some parameter $\theta \in \Theta$. Let $\Pi = \{\pi_x(\cdot):x \in \chi\}$ be a family of possible distributions (termed "inverse probability distributions") on $\Theta$ depending on $x \in \chi$. A theorem is given to characterize the admissibility of a decision rule $\delta$ which minimizes the expected loss with respect to the distribution $\pi_x(\cdot)$ for each $x \in \chi$. The theorem is partially extended to the case when the sample space and the parameter space are not necessarily finite. Finally a notion of "admissible consistency" is introduced and a necessary and sufficient condition for admissible consistency is provided when the parameter space is finite, while the sample space is countable.

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. nonnegative rv's with nondegenerate df $F$. Define $\tilde{N}(t) = {\tt\#}\{j: X_1 + \cdots + X_j \leq t\}$. In "testing with replacement" (also known as "renewal testing") $n$ independent copies of $\tilde{N}$ are observed each over the time interval $\lbrack 0, \tau \rbrack$ and we are interested in nonparametric estimation of $F$ based on these observations. We prove consistency of the product limit estimator as $n\rightarrow\infty$ for arbitrary $F$, and weak convergence in the case of integer valued $X_i$. We state the analogue of this result for continuous $F$ and briefly discuss the similarity of our results with those for the product limit estimator in the model of "random censorship."

Let $X_1, \cdots, X_n$ be i.i.d. random variables with probability distribution $F_{\theta, p}$ indexed by two real parameters. Let $\hat{p} = \hat{p}(X_1, \cdots, X_n)$ be an estimate of $p$ other than the maximum likelihood estimate, and let $\hat{\theta}$ be the solution of the likelihood equation $\partial/\partial \theta \ln L(\mathbf{x}, \theta, \hat{p}) = 0$ which maximizes the likelihood. We call $\hat{\theta}$ a pseudo maximum likelihood estimate of $\theta$, and give conditions under which $\hat{\theta}$ is consistent and asymptotically normal. Pseudo maximum likelihood estimation easily extends to $k$-parameter models, and is of interest in problems in which the likelihood surface is ill-behaved in higher dimensions but well-behaved in lower dimensions. We examine several signal-plus-noise, or convolution, models which exhibit such behavior and satisfy the regularity conditions of the asymptotic theory. For specific models, a numerical comparison of asymptotic variances suggests that a pseudo maximum likelihood estimate of the signal parameter is uniformly more efficient than estimators proposed previously.

The problem of estimating a normal mean has received much attention in recent years. If one assumes, however, that the true mean lies in a bounded interval, the problem changes drastically. In this paper we show that if the interval is small (approximately two standard deviations wide) then the Bayes rule against a two point prior is the unique minimax estimator under squared error loss. For somewhat wider intervals we also derive sufficient conditions for minimaxity of the Bayes rule against a three point prior.

In 1959 A. Renyi proposed a set of axioms for a measure of dependence for pairs of random variables. In the same year A. Sklar introduced the general notion of a copula. This is a function which links an $n$-dimensional distribution function to its one-dimensional margins and is itself a continuous distribution function on the unit $n$-cube, with uniform margins. We show that the copula of a pair of random variables $X, Y$ is invariant under a.s. strictly increasing transformations of $X$ and $Y$, and that any property of the joint distribution function of $X$ and $Y$ which is invariant under such transformations is solely a function of their copula. Exploiting these facts, we use copulas to define several natural nonparametric measures of dependence for pairs of random variables. We show that these measures satisfy reasonable modifications of Renyi's conditions and compare them to various known measures of dependence, e.g., the correlation coefficient and Spearman's $\rho$.

When a BIBD or a Group Divisible design with $\lambda_2 = \lambda_1 + 1$ is extended by certain disjoint and binary blocks the resulting structure is proved $E$-optimal. A BIBD abridged by a certain number of such blocks is also shown $E$-optimal. These optimality results hold over the class of all block designs (with the respective sets of parameters). Proofs rely mainly on averaging information matrices, which proves useful in many settings related to design optimality.

For a given class of linear models in standard form an optimal experimental design is to be chosen for estimating some linear functions of the unknown parameters. An optimality criterion is defined to be a real function of the covariance matrices of the Gauss-Markov estimators. Conditions which are imposed on the criteria are monotonicity, quasiconvexity or quasiconcavity, and invariance or order-invariance. A characterization of the $D$-criterion by order-invariance is included which strengthens a result of P. Whittle. In the main part of the paper optimal designs for the usual two-way layouts in ANOVA are computed for large classes of optimality criteria. Some related optimization problems are solved with the technique of majorization of vectors in the sense of Schur.

Asymptotic normality of linear combinations of order statistics of the form $T_n = n^{-1} \sum J(i/(n + 1))X_{in}$ is investigated along with a slightly trimmed version of $T_n$. Theorem 5 of Stigler (1974) is extended to show asymptotic normality of $T_n$ for a wide class of score functions. In addition, a proof of Theorem 4 of Stigler (1974) is given.

The problem of optimal design for linear regression extrapolation is discussed. A restriction is introduced which will ensure that the number of optimal design points is equal to the number of independent regressors.

Under a general Gauss-Markoff model $\{\mathbf{y}, \mathbf{X,\beta, V}\}$, a necessary and sufficient condition is established for a linear transformation, $\mathbf{F}$, of the observable random vector $\mathbf{y}$ to have the property that there exists a linear function of $\mathscr{Fy}$ which is a BLUE of $\mathbf{X\beta}$. A method for deriving a required BLUE from the transformed model $\{\mathbf{Fy, FX\beta, FVF}'\}$ is also indicated.

In this paper a necessary and sufficient condition for a predictor to be the best invariant predictor in location and scale families is given. Using this condition, it is shown that the best invariant predictor is expressed by a linear combination of the best unbiased predictor and the best unbiased estimator of the scale parameter.