Multivariate generalizations of the Wald-Wolfowitz runs statistic and the Smirnov maximum deviation statistic for the two-sample problem are presented. They are based on the minimal spanning tree of the pooled sample points. Some null distribution results are derived and a simulation study of power is reported.

The likelihood function is the common basis of all parametric inference. However, with the exception of an ad hoc definition by Fisher, there has been no such unifying basis for prediction of future events, given past observations. This article proposes a definition of predictive likelihood which can help to remove some nonuniqueness problems in sampling-theory predictive inference, and which can produce a simple prediction analog of the Bayesian parametric result, posterior $\propto$ prior $\times$ likelihood, in many situations.

A vector $\mu = (\mu_1, \cdots, \mu_n)$ is said to be upper [lower] starshaped if $\mu_{m + 1} \geqslant 0 \lbrack \leqslant \mu_{m + 1} \leqslant \bar{\mu}_m\rbrack m = 1, \cdots, n - 1$, where $\bar{\mu}_m$ is a weighted average of $\mu_1, \cdots, \mu_m$. Obtained is the maximum likelihood estimate of $\mu$ when the $\mu_i$'s are the means of $n$ Poisson or normal populations and $\mu$ is known to be starshaped. The method is applied to obtain estimators of IHRA (increasing hazard rate average) distribution functions.

During the last twenty years several results on the conditional properties of statistical procedures have appeared in the literature. However, apart from results linking conditional properties and Bayesianity, few general results have been found. Here a systematic set of definitions of conditional properties is given and a systematic investigation of their consequences is started. An analogy between interval and point estimation is used both to extend definitions of conditional properties to point estimation and to extend the usual definition of admissibility for point estimation to provide a new definition of admissibility for interval estimation.

A systematic investigation of the conditional properties of statistical procedures for location and scale parameters is commenced. These conditional properties are compared with the known admissibility properties of point estimators in the same situations.

Almost sure limit theorems are proved for maxima of functions of moving blocks of size $c \log n$ of independent rv's and for maxima of functions of the empirical probability measures of these blocks. It is assumed that for the functions considered a first-order large deviation statement holds. It is well known that the indices of these large deviations are, in most cases, expressible in terms of Kullback-Leibler information numbers, and the a.s. limits of the above maxima are the inverses of these indices evaluated at $1/c$. Several examples are presented as corollaries for frequently used test statistics and point estimators.

Statistics of the form $H_n = \sum d_{ijn}h_n(X_i, X_j)$ are considered, where $X_1, X_2, \cdots$, are independent and identically distributed random variables, the diagonal terms, $d_{iin}$, are equal to zero, and $h_n(x, y)$ is a symmetric real valued function. The asymptotic normality of such statistics is proven and the result then combined with work of Jogdeo on statistics that are weighted sums of bivariate functions of ranks to find sufficient conditions for asymptotic normality of permutation statistics derived from $H_n$.

It is shown that under fairly mild conditions linear combinations of independent normally distributed random variables with random coefficients tend to zero almost everywhere. The result is applied to parameter estimation in linear regression models.

Let $T \subseteq I$ be sets of real numbers. Let $\{Y(t): t\in I\}$ be a real time series whose covariance kernel is assumed known and positive definite. The mean is assumed either to be known or to be an unknown member of a known class of functions on $I$. For each fixed $s\in I, Y(s)$ is predicted by a minimum mean square error unbiased linear predictor $\hat{Y}(s)$ based on $\{Y(t): t\in T\}$. If $\hat{y}(s)$ is the evaluation of $\hat{Y}(s)$ given that the sample path for $\{Y(t): t\in T\}$ is an unknown element of a known collection of functions on $T$, then $\hat{y}(s)$ is a prediction for $Y(s)$ and the function $\hat{y}$ is called a prediction function. Mean-estimation functions are defined similarly. For certain prediction problems based on imprecise observations, characterizations are obtained for these functions in terms of the covariance structure of the process. For a particular prediction problem $\hat{y}$ is shown to be a spline function interpolating a convex set.

A repeated measures design occurs in analysis of variance when a particular individual receives several treatments. Let $X_i = (x_{il}, \cdots, x_{ip})'$ be the vector of observations on the $i$th individual. It is assumed that the $X_i$ are independently normally distributed with mean $\mu_i$ and common covariance $\sum > 0$. The researcher wants to test hypotheses about the $\mu_i$. Let $\varepsilon_i = (\varepsilon_{i1}, \cdots, \varepsilon_{ip})' = X_i - \mu_i$. For this paper, in order to get powerful tests, the simplifying assumption that the $\varepsilon_{i1}, \cdots, \varepsilon_{ip}$ are exchangeable is made. We assume that the design is given and use a coordinate-free approach to find optimal (i.e., UMP invariant, UMP unbiased, most stringent, etc.) procedures for testing a large class of hypotheses about the $\mu_i$.

The multivariate normal patterned mean and covariance matrix testing problem is studied for general one and $k$-population hypotheses. T. W. Anderson's iterative algorithm for finding the maximum likelihood estimates, the forms of the likelihood ratio tests, and asymptotic chi-square distributions of these tests under the null hypothesis are given. The nonnull asymptotic normal distribution is derived using the standard delta method. This derivation involves using several extensions of matrix identities given in Anderson, matrix derivatives and asymptotic likelihood equations. The forms of the variances are greatly simplified using a result of Szatrowski when the maximum likelihood estimates under the null hypothesis have explicit representations.

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqslant 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\sum$. We wish to estimate $\theta$ under the quadratic loss $L(\delta; \theta, \sum) = \lbrack\mathrm{tr}(Q\sum)\rbrack^{-1}(\delta - \theta)'Q(\delta - \theta)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta_{k, h}(X, W) = \lbrack I - kh(X'W^{-1}X)\lambda_1(QW/n^\ast)Q^{-1}W^{-1}\rbrack X,$ where $W : p \times p$ is observed independently of $X$ and has a Wishart distribution with $n$ degrees of freedom and parameter $\sum, \lambda_1(A)$ denotes the minimum characteristic root of $A$, and $h(t): \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is absolutely continuous with respect to Lebesgue measure, is nonincreasing, and satisfies the additional requirements that $th(t)$ is nondecreasing and $\sup_{t \geqslant 0}th(t) = 1$. With $h(t) = t^{-1}$, the class $\delta_{k, h}$ specializes to that considered by Berger, Bock, Brown, Casella and Gleser (1977). For the more general class considered in the present paper, it is shown that there is an interval $\lbrack 0, k_{n, p}\rbrack$ of values of $k$ (which may be degenerate for small values of $n - p)$ for which $\delta_{k, h}$ is minimax and dominates the usual estimator $\delta_0 \equiv X$ in risk.

The paper studies experiments which, for nonrandom stopping rules, resemble Koopman-Darmois families. It is shown that asymptotically one can limit oneself to sequential stopping rules which depend only on the terms entering in the Koopman-Darmois approximations, whether or not these terms are sums of independent variables. One can also obtain asymptotic results by studying similar problems on suitable processes with independent increments.

Ordinarily a Bayesian estimation procedure uses one prior distribution to obtain a unique estimation rule (its Bayes rule). From the decision theoretical point of view, this procedure can be regarded as a convenient way to obtain admissible decision rules. However, many intuitively appealing, admissible estimation rules cannot be obtained directly in this way. We propose a new mechanism, called the Stepwise Bayesian Procedure (SBP). When the parameter space contains only finitely-many points and the loss function is strictly convex, this SBP can be used to obtain every admissible estimation rule. A relationship between SBP and the limiting Bayes rules is given.

Let $X_1, \cdots, X_n$ be $n \geqslant 2$ positive random variables and $G(\mathbf{X})$ a positive variable satisfying $G(a\mathbf{X}) = aG(\mathbf{X})$ for all $a > 0$. Then $G$ is a size variable, and $\mathbf{X}/G$ is a shape vector. If $X_1, \cdots, X_n$ are independent, then the independence of shape and the size variable $G(\mathbf{X})$ characterizes (i) the lognormal distribution if $G(\mathbf{X}) = \Pi X^{1/n}_i$, (ii) the generalized gamma distribution if $G(\mathbf{X}) = (\sum X^b_i)^{1/b}$, (iii) the Pareto distribution or its discrete analogue if $G(\mathbf{X}) = \min(\mathbf{X})$, and (iv) the power-function distribution or its discrete analogue if $G(\mathbf{X}) = \max(\mathbf{X})$. It is shown here that if $X_1, \cdots, X_n$ have piecewise continuous density functions and $G$ is a continuous function then these four size variables are effectively the only ones for which such independence properties are attainable. A connection with the theory of sufficient statistics for a scale parameter is also considered.

Geary's characterization of the normal distribution asserts that if $n \geqslant 2$ i.i.d. observations come from some distribution on the line, then the sample mean and variance are independent if and only if the observations are normally distributed. A similar characterization is established here for the uniform distribution on the circle. Given a sample of $n \geqslant 2$ i.i.d. random angles from a distribution defined by a density on the circle satisfying some mild regularity conditions, the sample mean direction and resultant length are independent if and only if the angles come from the uniform distribution.

Asymptotically optimal (a.o.) empirical Bayes (EB) estimators are proposed. Speeds and the best possible speed at which these estimators are a.o. are investigated. The underlying component problem is the squared error loss estimation of $\theta$ based on an observation $X$ whose conditional (on $\theta$) $\operatorname{pdf}$ is of the form $u(x)C(\theta)\exp(\theta x)$. The function $u$ could have infinitely many discontinuities; $\theta$ is distributed according to an unknown and unspecified $G$ with support in $\Theta$, and $\Theta$ could be unbounded. Using $n$ independent past experiences of the component problem, EB estimators $\phi_n$ for the present problem are exhibited for each integer $r > 1$. The risks $R(\phi_n, G)$ due to $\phi_n$ are shown to converge to the minimum Bayes risk $R(G)$. In particular, for each $\delta$ in $\lbrack r^{-1}, 1\rbrack$, sufficient conditions are given under which $c_1n^{-2(r - 1)/(1 + 2r)} \leqslant R(\phi_n, G) - R(G) \leqslant c_2n^{-2(\delta r - 1)/(1 + 2r)}$, where $c_1$ and $c_2$ are positive constants. The right hand-side inequality holds uniformly in $G$ satisfying certain conditions, while the other holds at all degenerate $G$ and for all large $n$. (Thus with $\delta$ close to one, $\phi_n$ achieves almost the exact rate.) Examples of exponential families such as normal, gamma and one with $\operatorname{pdf's}$ having infinitely many discontinuities are given where the conditions for the above inequalities are satisfied uniformly in $G$ with $\int|\theta|^{2r\delta}dG(\theta) < \infty$.

Mill's ratio is expressed as a convergent series in orthogonal polynomials. Truncation of the series provides an approximation for the complemented normal distribution function $Q(x)$, with its maximum error at a finite value of $x$. The analogous approximation for $xQ(x)$ is used to obtain a new method of calculating the bivariate normal probability function.

Systems with exponential life formed from independent IFRA components are studied. It is shown that (a) in the case of monotonic systems, the system must be essentially a series system of exponential components and (b) in the case of systems whose life is the sum of component lives, all but one of the components are degenerate at zero while the remaining one is exponential.

The problem of partitioning a set of normal populations by their locations with respect to a control was investigated by Tong, and Seeger considered an extension of this to a comparison with 2 controls. This note points out that the main result in the latter paper is in error, and provides a table for use in the application of Seeger's procedure.

The weak convergence of the Kaplan-Meier estimate for censored survival data to a Gaussian process is used to construct asymptotic confidence bands for the survival curves. The method involves transforming to obtain Brownian motion and using straight line boundaries and hitting probabilities for Brownian motion.