A random variable $X$ is said to have distribution in the class $\mathscr{E}_0$ if, for some real valued, positive function $a(\bullet)$, the identity $E\{(X - \mu)g(X)\} = E\{a(X)g'(X)\}$ holds for any absolutely continuous real valued function $g(\bullet)$ satisfying $E|a(X)g'(X)| < \infty$. Examples of a random variable $X$ possessing a distribution in $\mathscr{E}_0$ are (i) $X$ normally distributed with mean $\mu$ and standard deviation 1, (ii) $X$ having a gamma density with mean $\mu$ and location parameter 1, (iii) $X = 1/Y$ where $Y \sim \lbrack(n - 2)\rbrack^{-1}\chi_n^2, n > 2$. Suppose $X_1,\cdots, X_p, p \geqq 3$, are independently distributed with distributions in $\mathscr{E}_0$, for some function $a(\bullet)$, and with means $\mu_1,\cdots, \mu_p$. Define $b(x) = \int a(x)^{-1} dx$, where the integral is interpreted as indefinite, $B_i = b(X_i), S = \sum^p_{i=1} B_i^2, X' = (X_1,\cdots, X_p)$ and $B' = (B_1,\cdots, B_p)$. Then the estimator $X - ((p - 2)/S)B$ dominates $X$ if sum of squared error loss is assumed. Similar estimators are obtained, when $p \geqq 4$, which shrink towards an origin determined by the data. There are corresponding results for some discrete exponential families.

This paper considers likelihood ratio tests for testing hypotheses that a collection of parameters satisfy some order restriction. The first problem considered is to test a hypothesis specifying an order restriction on a collection of means of normal distributions. Equality of the means is the subhypothesis of the null hypothesis which yields the largest type I error probability (i.e., is least favorable). Furthermore, the distribution of $T = -\ln$ (likelihood ratio) is similar to that of a likelihood ratio statistic for testing the equality of a set of ordered normal means. The least favorable status of homogeneity is a consequence of a result that if $X$ is a point and $A$ a closed convex cone in a Hilbert space and if $Z \in A$, then the distance from $X + Z$ to $A$ is no larger than the distance from $X$ to $A$. The results of a Monte Carlo study of the power of the likelihood ratio statistic are discussed. The distribution of $T$ is also shown to serve as the asymptotic distribution for likelihood ratio statistics for testing trend when the sampled distributions belong to an exponential family. An application of this result is given for underlying Poisson distributions.

In the discrimination problem the random variable $\theta$, known to take values in $\{1,\cdots, M\}$, is estimated from the random vector $X$. All that is known about the joint distribution of $(X, \theta)$ is that which can be inferred from a sample $(X_1, \theta_1),\cdots, (X_n, \theta_n)$ of size $n$ drawn from that distribution. A discrimination rule is any procedure which determines a decision $\hat{\theta}$ for $\theta$ from $X$ and $(X_1, \theta_1),\cdots, (X_n, \theta_n)$. A rule is called $k$-local if the decision $\hat{\theta}$ depends only on $X$ and the pairs $(X_i, \theta_i)$ for which $X_i$ is one of the $k$-closest to $X$ from $X_1,\cdots, X_n$. It is shown that for any $k$-local discrimination rule, the mean-square difference between the probability of error for the rule and its deleted estimate is bounded by $A/n$ where $A$ is an explicitly given small constant which depends only on $M$ and $k$. Thus distribution-free confidence intervals can be placed about probability of error estimates for $k$-local discrimination rules.

We study a class of decision rules based on an adaptive partitioning of an Euclidean observation space. The class of partitions has a computationally attractive form, and the related decision rule is invariant under strictly monotone transformations of coordinate axes. We provide sufficient conditions that a sequence of decision rules be asymptotically Bayes risk efficient as sample size increases. The sufficient conditions involve no regularity assumptions on the underlying parent distributions.

Nonparametric estimators are proposed for transition probabilities in partial Markov chains relative to multiple decrement models. The estimators are generalizations of the product limit estimator. We study the bias of the estimators, prove a strong consistency result and derive asymptotic normality of the estimators considered as stochastic processes. We also compute their efficiency relative to the maximum likelihood estimators in the case of constant forces of transition.

Bahadur and Raghavachari have formulated the likelihood ratio method of finding the exact slope of a sequence of test statistics which does not require explicit estimation of large deviation probabilities. The method is described herein, and readily verifiable conditions under which it may be invoked are given. The method is then used to find the exact slopes of certain sequences of test statistics arising in multivariate analysis.

An extremal problem in large deviation is solved for the one and two sample $t$-statistic. Let $T_n = \bar{X}/s_X$ be the $t$-statistic, except for normalizing constants, based on $n$ independent observations fron $F$ where $\bar{X}$ and $s_X$ are the sample mean and standard deviation. The statistic for the two sample case is $T_n = (\bar{X} - \bar{Y})/\lbrack(s^2_X + s^2_Y)/2\rbrack^\frac{1}{2}$ based on two independent samples of size $n$. Let $a$ be positive. We find the rate of convergence to 0 of $\sup_F P_F(T_n \geqq a)$ where the $\sup$ is taken over $F$'s symmetric at 0 in the one sample case and all $F$'s in the two sample case. The results are applied to obtain the Bahadur exact slopes of the $t$-statistic for the full nonparametric hypothesis testing problems. We include a table giving the slopes of the $t$-statistics, the slopes of standard rank statistics and the maximum possible slopes. This table shows that, in both the one and two sample situations, the $t$-statistic has, as expected, efficiency strictly less than one when compared with the normal scores statistic, and the slope of the normal scores test is very close to the maximum possible slope, at normal alternatives.

The notion of Bahadur efficiency for test statistics is extended to the sequential case and illustrated in the specific context of testing one-sided hypotheses about a normal mean. An analog of Bahadur's theorem on the asymptotic optimality of the likelihood ratio statistic is seen to hold in the normal case. Some possible definitions of attained level for a sequential experiment are considered.

Interval estimation of variance components is studied for the unbalanced one-way random effects model. An easily calculated function, $W$, of the harmonic mean of the class sizes and of the sample variance of the class means is found to be important. The exact distribution of $W$ is found and is shown to be excellently approximated by a chi-square distribution. The random variable $W$ is used to construct interval estimates for (i) the between classes variance component and (ii) the ratio of the variance components and thus for the intraclass correlation and heritability. For most one-way unbalanced designs use of these approximate interval estimators will work very well.

Asymptotic normality is established for multivariate linear rank statistics of general type in the non-i.i.d. case covering null hypotheses as well as almost arbitrary alternatives. The functions generating the regression constants and the scores are allowed to have a finite number of discontinuities of the first kind, and to tend to infinity near 0 and 1. The proof is based on properties of empirical df's in the non-i.i.d. case and is patterned on the 1958 Chernoff-Savage method. As special cases e.g. rank statistics used for testing against regression and rank statistics for testing independence are included.

In the simple linear regression problem $\{Y_i = \alpha + \beta x_i + e_i i = 1,\cdots, n, e_i$ i.i.d. $\sim F$ continuous, $x_1 \leqq \cdots \leqq x_n$ known, $\alpha, \beta$ unknown$\}$ we investigate the following type of estimator: To each $s_{ij} = (Y_j - Y_i)/(x_j - x_i)$ with $x_i < x_j$ attach weight $w_{ij}$ and as estimator for $\beta$ consider the median of this weight distribution over the $s_{ij}$. A confidence interval for $\beta$ is found by taking certain quantiles of this weight distribution. The asymptotic behavior of both is investigated and conditions for optimal weights are given.

This paper considers the estimation of a location parameter $\theta$ in a one-sample problem. The asymptotic performance of a sequence of estimates $\{T_n\}$ is measured by the exponential rate of convergence to 0 of $\max \{P_\theta(T_n < \theta - a), P_\theta(T_n > \theta + a)\}, \text{say} e(a).$ This measure of asymptotic performance is equivalent to one considered by Bahadur (1967). The optimal value of $e(a)$ is given for translation invariant estimates. Some computational methods are reviewed for determining $e(a)$ for a general class of estimates which includes $M$-estimates, rank estimates and Hodges-Lehmann estimates. Finally, some numerical work is presented on the asymptotic efficiencies of some standard estimates of location for normal, logistic and double exponential models.

Repeated measurements designs are concerned with scientific experiments in which each experimental unit is assigned more than once to a treatment, either different or identical. It is shown that a family of balanced repeated measurements designs which are very popular among experimenters are universally optimal in a relatively large class of competing designs.

An estimation procedure for stochastic processes based on the minimization of a sum of squared deviations about conditional expectations is developed. Strong consistency, asymptotic joint normality and an iterated logarithm rate of convergence are shown to hold for the estimators under a variety of conditions. Special attention is given to the widely studied cases of stationary ergodic processes and Markov processes with are asymptotically stationary and ergodic. The estimators and their limiting covariance matrix are worked out in detail for a subcritical branching process with immigration. A brief Monte Carlo study of the performance of the estimators is presented.

It is known that the autocorrelation function of a stationary discrete-time scalar process can be uniquely characterized by the so-called partial autocorrelation function, which is a sequence of numbers less or equal to one in magnitude. We show here that the matrix covariance function of a multivariate stationary process can be characterized by a sequence of matrix partial correlations, having singular values less than or equal to one in magnitude. This characterization can be used to extend to the multivariate case the so-called maximum entropy spectral analysis method.

A set of recurrences is developed for computing the probability content of any prior or posterior confidence bands for a distribution function assuming the parameter of the prior Dirichlet process known. When the prior parameter is unknown and is estimated from the sample, nonparametric estimates of the probability content of the posterior bands are obtained.

An adaptive approach is considered as an alternative to the classical indifference-zone formulation of the problems of ranking and selection. With a fixed $\gamma^\ast$, the proposed procedure calls for the termination of sampling when the estimated probability of correct selection exceeds $\gamma^\ast$ for the first time. Asymptotic properties of this procedure are proved as $\gamma^\ast \rightarrow 1$, and Monte Carlo results show that the procedure is well behaved even for moderate $\gamma^\ast$. Since the stopping variables depend on the estimators of the ordered parameters, distributions of the estimators as functions of the parameters are carefully studied via majorization.

A general method of constructing procedures which are both admissible and asymptotically optimal in finite state sequence compound decision problems is suggested and applied to the situation of a two state classification component. When used in an empirical Bayes setting, procedures so constructed are seen to be both admissible and asymptotically optimal.

Convergence with probability one of a recursive stochastic approximation algorithm is considered. Some extensions of previous results for the Robbins-Monro and the Kiefer-Wolfowitz procedures are given. An inportant feature of the approach taken here is that the convergence analysis can be directly extended to more complex algorithms.

Wynn gives an example of a balanced incomplete block design for 24 varieties in 56 blocks of three plots each which involves only 24 distinct blocks. It is shown that this design can be obtained from a hierarchic group divisible association scheme, and is one of a set of eight possible designs.