This is a survey of recent work on Edgeworth expansions for $(M)$ estimates, rank tests and some other statistics arising in nonparametric models. A Berry-Esseen theorem for $U$-statistics which seems to be new is also proved.

Techniques for improving on equivariant estimators are described. They may be applied, although without assurance of success, whatever be the family of underlying distributions. The loss function is required to satisfy an intuitively reasonable condition but is otherwise arbitrary. One of these techniques amounts to a sample space, orbit-by-orbit analysis of the conditional expected loss given the orbit. It yields, when successful, a "testimator". A second technique obtains the limit of a certain sequence of "testimator-like" estimators. The result is "smoother" than a testimator and often identical to a generalized Bayes estimator over much of its domain. Applications are presented. In the first we extend results of Stein (1964) and obtain a minimax estimator which is generalized Bayes, and in a univariate subcase, admissible within the class of scale-equivariant estimators. In the second, we extend a result of Srivastava and Bancroft (1967).

Upper bounds are established for the probability that, in sampling without replacement from a finite population, the sample sum exceeds its expected value by a specified amount. These are obtained as corollaries of two main results. Firstly, a useful upper bound is derived for the moment generating function of the sum, leading to an exponential probability inequality and related moment inequalities. Secondly, maximal inequalities are obtained, extending Kolmogorov's inequality and the Hajek-Renyi inequality. Compared to sampling with replacement, the results incorporate sharpenings reflecting the influence of the sampling fraction, $n/N$, where $n$ denotes the sample size and $N$ the population size. We go somewhat beyond previous work by Hoeffding (1963) and Sen (1970). As in the latter reference, martingale techniques are exploited. Applications to simple linear rank statistics are noted, dealing with the two-sample Wilcoxon statistic as an example. Finally, the question of sharpness of the exponential bounds is considered.

Analogous to the Donsker theorem on partial cumulative sums of independent random variables, for a broad class of one-sample rank order statistics, weak convergence to Brownian motion processes is studied here. A simple proof of the asymptotic normality of these statistics for random sample sizes is also presented. Some asymptotic results on renewal theory for one-sample rank order statistics are derived.

This paper describes a new construction of uniformly asymptotically efficient rank estimates in the one and two-sample location models. The method adopted differs from van Eeden's (1970) earlier construction in three respects. First, the whole sample, rather than a vanishingly small fraction of the sample, is used in estimating the efficient score function. Secondly, a Fourier series estimator is used for the score function rather than a window estimator. Thirdly, the linearized rank estimates corresponding to the estimated score function provide the uniformly asymptotically efficient location estimates. These estimates are asymptotically efficient over a larger class of distributions than the van Eeden estimates and should approach their asymptotic behavior more rapidly.

We shall consider the hypothesis of randomness under which two samples $X_1, \cdots, X_n$ and $Y_1, \cdots, Y_m$ have an identical but arbitrary continuous distribution. The vector of ranks $(R_1, \cdots, R_{n+m})$ will be shown to be asymptotically sufficient in the Bahadur sense for testing randomness against a general class of two-sample alternatives, simple ones as well as composite ones. In other words, the best exact slope will be attainable by rank statistics, uniformly throughout the alternative.

The multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution. Such distributions are shown equivalent to the more strongly structured multivariate exponential distributions described by Marshall and Olkin in 1967 in the sense that a multivariate exponential distribution can be found that gives the same marginal distribution for each minimum. The basic application of the result is that in computing the reliability of a coherent system a joint distribution for the component life lengths with exponential minimums can be replaced by a multivariate exponential distribution. It follows that the life length of the system has an increasing hazard rate average distribution. Other applications include characterizations of multivariate exponential distributions and the derivation of a positive dependence condition for multivariate distributions with exponential minimums.

If $x(\bullet)$ is a time series which may be written as $x(t) = s(t) + n(t)$ where $t$ is an integer, $s(\bullet)$ an autoregressive signal of order $q$ and $n(\bullet)$ white noise, then the model has $q + 2$ parameters. These are (i) the $q$ autoregressive parameters (ii) the residual variance of the autoregressive scheme and (iii) the variance of the white noise. A method is proposed to estimate the $q + 2$ parameters. This method is based on analogies with regression theory and in the case of a normal series yields strongly consistent efficient estimators.

In this paper, asymptotic expansions of the non-null distributions of the likelihood ratio criteria are obtained for testing the hypotheses: (a) $H_1: \Sigma = \sigma^2I, \sigma^2 > 0$, (b) $H_2: \Sigma_1 = \Sigma_2$, against alternatives which are close to the hypothesis. These expansions are of chi-square type. The first problem has been considered by Sugiura (1969) but because of the singularity at the hypothesis, his expansion will not be good for alternatives close to the hypothesis. The second problem has been considered by de Waal (1970) but the expansion given by him is invalid.

This paper is concerned with rates of convergence in the sequence-compound decision problem when the component problem is the squared-distance loss estimation of the mean of $m$-variate normal distribution with covariance matrix I. Section 1 introduces some notation and discusses the earlier work related to this problem. In Section 2, we prove two lemmas which are required in later sections. Sections 3, 4 and 5 exhibit sequence-compound decision procedures whose modified regrets are $O(n^{-1/m+4})$, near $O(n^{-\frac{1}{4}})$ and near $O(n^{-\frac{1}{2}})$ respectively, all the orders being uniform in parameter sequences concerned. In Section 6, comparisons have been made between the procedures given in Sections 3, 4 and 5. We conclude the paper with a few remarks.

In a continuous production process, samples of fixed size are taken at regular intervals of time and a statistic $X_n$ is computed from the $n$th sample, $n = 1,2, \cdots$. In this paper, we study process inspection schemes which stop the production and take corrective action with $N = \text{first} n \geqq 1$ such that $\sum^n_{i=1} c_{n-i} X_i \geqq h$, where $h$ is a preassigned constant and $c_0 \geqq c_1 \geqq \cdots \geqq c_{k-1} > 0 = c_k = c_{k+1} = \cdots$ is a suitably chosen sequence of weights. The average run length of such procedures is examined, and in the normal case, numerical comparisons with the average run length of the usual Shewhart Chart are given. In connection with the normal case, the first passage times of more general Gaussian sequences are studied and an asymptotic theorem is obtained. The first passage time $N$ for more general weighted sums, where the sequence $(c_n)$ is not assumed to be eventually zero but is assumed to be at least square summable, is also considered.

Let $D$ be a BIB $(\nu, b, r, k, \lambda)$ design on $\Omega$. Let also $L \subset \Omega$ with cardinality of $L$ being $n \leqq \nu - 2$. We define $D$ to be locally resistant of degree $n$ if upon deletion of all the experimental units in $D$ assigned to the treatments in $L$ the remaining structure is variance balanced in the sense that under the usual homoscedastic additive linear model every normalized estimable linear function of the treatment effects are estimable with the same variance. $D$ is defined to be globally resistant of degree $n$ if it has the above property with respect to any subset $L \subset \Omega$ as long as its cardinality is $n. D$ is said to be susceptible if it is not resistant to any nonempty set $L$. Application of these concepts in various branches of sciences and engineering has been indicated. In this paper we have characterized all locally and globally resistant designs of degree one in two different ways. Through one of these characterizations we have been able to relate our theory to the theory of $t$-designs or tactical configurations. Methods for constructing some families of locally and globally resistant designs of degree one are provided. We have also shown that the property of being resistant depends not only on the parameters of $D$ but also depends on the way $D$ has been constructed. To illustrate this we have given three BIB (10, 30, 12, 4, 4) designs; the first design is susceptible, the second is locally resistant to the deletion of a single treatment and the third design is globally resistant. We have also indicated that every BIB $(\nu, b, r, k, \lambda)$ design is locally resistant of degree $k$ if $b = \nu$. Several miscellaneous results are also given, among which a locally resistant design of degree two is included. Several unsolved problems are indicated in the final section.

For continuous time Markov decision chains of finite duration, we show that the vector of maximal total rewards, less a linear average-return term, converges as the duration $t \rightarrow \infty$. We then show that there are policies which are both simultaneously $\varepsilon$-optimal for all durations $t$ and are stationary except possibly for a final, finite segment. Further, the length of this final segment depends on $\varepsilon$, but not on $t$ for large enough $t$, while the initial stationary part of the policy is independent of both $\varepsilon$ and $t$.

Simple expressions are given for the probability of a correct decision for the Anderson sequential procedure with a triangular boundary, for some values of parameters. This can be used to improve bounds of the probability of correct decision for the Paulson sequential ranking procedures and change slightly the comparison of these with the Bechhofer-Kiefer-Sobel procedures.

Robustness properties of the two-sample Wilcoxon test are established when the assumption of independence between samples is weakened to allow pairing. In particular, the test is shown to be (asymptotically) conservative when the bivariate distribution governing the pairing is positively quadrant dependent.

The paper gives an expression suitable for numerical computation and recursive relationships for the expected volume of the intersection of an $n$ dimensional sure sphere and an $n$ dimensional random sphere with a different radius, whose center follows a spherical distribution, the center of which does not coincide with that of the sure sphere. Several expressions are given for the distribution of the squared noncentral radial error when the vector observation follows a spherical distribution--a generalization of the noncentral chi-square distribution. Applications to specific spherical models are presented.

The problem of estimating powers of the variance of a normal distribution is considered when loss is essentially squared error. A class of minimax estimators is found by extending the techniques of Stein. It is shown, at least for estimating the variance, that a subclass of the above consists of generalized Bayes estimators.

When an admissible symmetric compound decision rule is applied to a sequence of simple hypothesis testing problems, the decisions are shown to exactly reflect the ordering of the component likelihood ratios. This leads to a characterization of admissible procedures which is closely related to the method ordinarily used in constructing compound decision rules. The extension to estimation problems is indicated.

Two different definitions of hyperadmissibility are considered. The relation between hyperadmissibility and admissibility is discussed. Special attention is given to the hyperadmissibility of the Horvitz-Thompson estimator.

It is proved that in a sequential design of a regression problem the variance of the 1.s. estimate for the response surface in the $n$th sequential point tends to zero with $n \rightarrow \infty$. This allows the proof of the convergence of certain procedures for computing $D_s$-optimum designs.

A recent result of Blackwell states that in a positive dynamic programming problem with countable state space, if there is an optimal policy, then there is a stationary optimal policy. We extend this result by allowing the state space to be Borel and by proving that if there is an optimal policy, then for any probability measure $\mu$ on the state space there is a stationary policy which is optimal on a set of $\mu$ measure 1.