Methods of generating prior distributions on spaces of probability measures for use in Bayesian nonparametric inference are reviewed with special emphasis on the Dirichlet processes, the tailfree processes, and processes neutral to the right. Some applications are given.

Let $\{X_j; j = 1, 2, \cdots\}$ be independent identically distributed random variables whose individual distribution $p_\theta$ is indexed by a parameter $\theta$ in a set $\Theta$. For two integers $m < n$ the experiment $\mathscr{E}_n$ which consists in observing the first $n$ variables is more informative than $\mathscr{E}_m$. Two measures of the supplementary information are described. One is the deficiency $\delta (\mathscr{E}_m, \mathscr{E}_n)$ introduced by this author. Another is a number $\eta(\mathscr{E}_m, \mathscr{E}_n)$ called "insufficiency" and related to previous arguments of Wald (1943). Relations between $\delta$ and $\eta$ are described. One defines a dimensionality coefficient $D$ for $\Theta$ and obtains a bound of the type $\eta(\mathscr{E}_m, \mathscr{E}_n) \leqq \lbrack 2D(n - m)/n\rbrack^{\frac{1}{2}}.$ Examples show that $\delta(\mathscr{E}_m, \mathscr{E}_n)$ may stay bounded away from zero in infinite dimensional cases, even if $m \rightarrow \infty$ and $n = m + 1$.

This paper proposes and develops a method for selecting a design to estimate a set of linear parametric functions in cases wherein the adequacy of the preliminary linear model is in doubt. The proposed method relies on the norm of the aliasing matrix. If the expected value of the estimator $\hat{\psi}$ of a set of linear functions $\psi = L_1\theta_1$ using a design $\Gamma$, under the true model is $E\lbrack\hat{\psi}\rbrack = \psi + A_\Gamma \theta_2$, then the norm $\|A_\Gamma\| =$ (trace $A_\Gamma' A_\Gamma)^{\frac{1}{2}}$ is presented as a measure for use in determining "alias balance" and "alias goodness." Therefore, $\|A_\Gamma\|$ may be used in the selection of a design for experimentation, and its behaviour under various operations is discussed. Some theorems concerning aliases of rank equivalent and complementary designs in certain settings are obtained.

General theorems on asymptotic normality of randomly trimmed and Winsorized means are obtained. A new small sample studentization is proposed. Many examples are presented. The two sample problem is also considered.

This paper considers linear functions of order statistics of the form $S_n = n^{-1} \sum J(i/(n + 1))X_{(i)}$. The main results are that $S_n$ is asymptotically normal if the second moment of the population is finite and $J$ is bounded and continuous a.e. $F^{-1}$, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if $J$ trims the extremes. In addition, asymptotic formulas for the mean and variance of $S_n$ are given for both the identically and non-identically distributed cases. All of the theorems of this paper apply to discrete populations, continuous populations, and grouped data, and the conditions on $J$ are easily checked (and are satisfied by most robust statistics of the form $S_n$). Finally, a number of applications are given, including the trimmed mean and Gini's mean difference, and an example is presented which shows that $S_n$ may not be asymptotically normal if $J$ is discontinuous.

Given $n$ independent observations with common density $f(x - \theta)$, and a rv $z$ independent of these with density $g(x - \theta) (f, g$ known except for $\theta$) a prediction region for $z$ is required. It is shown that the best translation invariant interval is optimal in two senses: (1) there is no other region with the same expected coverage (coverage is the probability of containing $z$) and uniformly smaller expected size (Lebesgue measure); (2) no other interval having the same confidence that the coverage exceeds $\beta$ (given) can have uniformly smaller expected length. The best invariant interval in each case is found, and the normal case is studied. The usual interval centered at $\bar{X}$ is not always optimal in the second sense if $\beta$ and/or confidence are small. A criterion involving expected coverage and the confidence of exceeding coverage $\beta$ is also examined. Again restrictions on these are needed for the usual normal interval to be optimal.

Berk (1970), LeCam (1953) and others have given conditions for the consistency of posterior distributions from a sequence of random variables. They have required that the sequence be i.i.d. We show that their results, Berk's in particular, may be extended to the general linear hypothesis with normal errors model (where the sequence of observations of the dependent variable need not be i.i.d.). We do not assume that the distribution governing the sequence of dependent variables has a regression function which satisfies the assumed model nor do we assume its errors are normal. Consistency is shown for both fixed and random sampling designs. We show that the convergence is to a projection of only the true regression function upon the space of regression functions given by the model. Finally, we assume that several such models are under consideration, each with a prior probability. We determine conditions for the a.s. convergence of their posterior probabilities to a degenerate distribution. Not all these conditions may be derived by any simple extension of Berk's results.

The paper gives convergence theorems for several sequential Monte-Carlo or stochastic approximation algorithms for finding a local minimum of a function $f(\bullet)$ on a set $C$ defined by $C = \{x: q^i(x) \leqq 0, i = 1, 2, \cdots, s\}. f(\bullet)$ is unknown, but "noise perturbed" values can be observed at any desired parameter $x \in C$. The algorithms generate a sequence of random variables $\{X_n\}$ such that (for a.a. $\omega$) any convergent subsequence of $\{X_n(\omega)\}$ converges to a point where a certain necessary condition for constrained optimality holds. The techniques are drawn from both stochastic approximation, and non-linear programming.

A zero-sum two person game is repeatedly played. Some of the payoffs are "absorbing" in the sense that, once any of them is reached, all future payoffs remain unchanged. Let $v_n$ denote the value of the $n$-times repeated game, and let $v_\infty$ denote the value of the infinitely-repeated game. It is shown that $\lim v_n$ always exists. When the information structure is symmetric, $v_\infty$ also exists and $v_\infty = \lim v_n$.

An analysis of the behavior of the $\operatorname{cdf}$ randomly determined by the subsample means yields results on the asymptotic relative efficiency and consistency of Hartigan's subsampling procedure. Examination of higher order approximation to the normal is carried out in two special cases.

Let $X$ and $Y$ be two $p \times 1$ random vectors distributed according to a normal distribution with respective mean vectors $\mu$ and $a\mu$ and covariance matrix $\begin{pmatrix}I_p & \rho I_p \\ \rho I_p & I_p\end{pmatrix}.$ Let $S$ be a random $p \times p$ matrix distributed as the Wishart distribution $W_p(I_p, r)$, independently of $X$ and $Y$. For fixed $a, \rho$, and $c$, some sufficient conditions are obtained for which $P\lbrack X'Y < c\rbrack$ and $P\lbrack X'S^{-1}Y < c\rbrack$ increase with $\mu'\mu$. These results are used to show a monotonicity property of the probabilities of correct classification of a class of rules for classifying an observation into one of two normal distributions. For the classification problem, some estimates of the probability of correct classification of the minimum distance rule are studied.

We have a population composed of two subpopulations whose probability properties are described by known univariate distribution functions, $G(x)$ and $H(x)$, respectively. The probability of observing an individual from the first population is $\theta$, from the second is $1 - \theta$. We assume $\theta$ is a random variable with a prior distribution on (0, 1) and find the Bayes rule for classifying $n$ observations as from $G$ or from $H$ when the loss function is equal to the number of misclassifications. The main results in the paper give the asymptotic properties of the Bayes rule and several proposed approximations.

We feel that a criterion for selecting optimal decision rules in a statistical decision problem should be selected rationally. Specifically, we would like a criterion to satisfy eight properties which we have given. It is known that none of the commonly used criteria satisfy these eight properties. We give sufficient conditions on the decision problem so that the iterated minimax regret criterion does satisfy all eight properties, and give examples to show that these sufficient conditions cannot be removed.

Spline functions, which are solutions to certain deterministic optimization problems, can also be regarded as solutions to certain stochastic optimization problems; in particular, certain linear least-squares estimation problems. Such an interpretation leads to simple recursive algorithms for interpolating and smoothing splines. These algorithms compute the spline using one data point at a time, and are useful in real-time calculations when data are acquired sequentially.

For each $n, X_n(1), \cdots, X_n(n)$ are independent and identically distributed continuous random variables over $(0, 1)$, with common density function equal to $1 + r(x)/n^{\frac{1}{2}}, r(x)$ unknown but satisfying certain regularity conditions. The problem is to test the hypothesis that $r(x) = 0$ for all $x$ in (0, 1). $Y_n(1) < \cdots < Y_n(n)$ are the ordered values of $X_n(1), \cdots, X_n(n). \delta$ is a fixed value in the open interval $(\frac{3}{4}, 1)$. It is shown that $Y_n(\lbrack n^\delta\rbrack), Y_n(2\lbrack n^\delta\rbrack), \cdots$ are asymptotically sufficient, and can be assumed to have a joint normal distribution for all asymptotic purposes. Using these facts, a test of the hypothesis is constructed with a good asymptotic power curve.

Under general circumstances it is shown that the sample autocovariances of a discrete, stationary, ergodic process with finite covariance which is also purely nondeterministic converge, uniformly on the lag, almost surely to the true values. The result is used to prove the almost sure convergence, uniform in a parameter, of an expression relevant to the estimation of a lagged relation between two series.

The purpose of this short paper is to point out how a slight modification of the formulation of the problem in Brown (1966) yields a variety of admissibility results for problems involving an unknown location or scale parameter. The results sketched, but not proved, include some applications of the analogous two parameter results in Brown and Fox (1974). The applications involve testing problems, estimation problems, and confidence set problems. We treat here only fixed sample size situations; although there also exist similar sequential applications. The proof is essentially that contained in Brown (1966).

A law of the iterated logarithm is proved for sample $p$-quantiles when the probability distribution function varies regularly at $\xi$ with $F(\xi) = p$.

The vague statement that "sampling with and without replacement from a finite population are approximately equivalent when the sampling fraction is small" is given a precise meaning in terms of limit theorems for distributions in $R^\infty$ and $D\lbrack 0, \infty)$.

Admissibility properties of the M.L.E. for the parameters of $m$ independent binomial distributions (when these parameters are known to be ordered) are determined for certain convex loss functions. It is shown that, except in two special cases, the M.L.E. is inadmissible whenever the total sample size is 7 or more.

Brunk studied integral regression functions and has obtained strong laws and limiting distributions for estimators of these functions. In this note we will study additional conditions that ensure a rate of convergence of the distribution function of the maximum absolute difference of an integral regression function and its estimator, suitably normalized, to the distribution function of a normalized maximum absolute value of partial sums of random variables. These results are corollaries of convergence results obtained by Sawyer and Rosenkrantz.

This paper provides proof of conjectures about the expected values of the $j$th elementary symmetric function of the roots (i.e. $E(\operatorname{tr}_j A)$) of a non-central Wishart matrix. Utilizing this result we have obtained computationally convenient formula for the expected value of the $j$th elementary symmetric function of the roots of central positive definite quadratic form in matrix argument.

Dwyer's (1937) $h$-statistic is extended to the generalized $h$-statistic $h_{p_1\cdots p_u}$ such that $E(h_{p_1\cdots p_u}) = \mu_{p_1} \cdots \mu_{p_u}$, similar to the extension of Fisher's $k$-statistic to the generalized $k$-statistic $k_{p_1\cdots p_u}$ requiring $E(k_{p_1\cdots p_u}) = \kappa_{p_1} \cdots \kappa_{p_u}$. The $h$-statistics follow simpler multiplication rules than for $k$-statistics and involve smaller coefficients. Generalized $h$-statistics are studied in terms of symmetric means, unrestricted sums, and ordered partitions, and their relationships with generalized $k$-statistics are established. The statistics are useful in obtaining approximate forms for sampling distributions when parent population is not completely known.

Many authors have studied the problem of obtaining an optimal plan for a dynamic programming problem or a stochastic game. In several of these works a maximal selection theorem of Dubins and Savage for u.s.c. (upper-semi-continuous) maps has played a key role. Our purpose is to relax the requirements that the maps be u.s.c. and still obtain maximal or at least nearly maximal selections.