Measures of location differentiable at every density in the Hellinger metric are constructed in this paper. Differentiability entitles these location functionals to the label "robust," even though their influence curves need not be bounded and continuous. The latter properties are, in fact, associated with functionals differentiable in the Prokhorov metric. A Hellinger metric concept of minimax robustness of a location measure at a density shape $f$ is developed. Asymptotically optimal estimators are found for minimax robust location measures. Since, at $f$, their asymptotic variance equals the reciprocal of Fisher information, asymptotic efficiency at $f$ and robustness near $f$ prove compatible.

This paper defines and studies for independent identically distributed observations a new parametric estimation procedure which is asymptotically efficient under a specified regular parametric family of densities and is minimax robust in a small Hellinger metric neighborhood of the given family. Associated with the estimator is a goodness-of-fit statistic which assesses the adequacy of the chosen parametric model. The fitting of a normal location-scale model by the new procedure is exhibited numerically on clear and on contaminated data.

Let $\hat{\mathbf{\Delta}}_M$ be an $M$-estimator (maximum-likelihood type estimator) and $\hat{\mathbf{\Delta}}_R$ be an $R$-estimator (rank estimator) of the parameter $\mathbf{\Delta} = (\Delta_1,\cdots, \Delta_p)$ in the linear regression model $X_{Ni} = \sum^p_{j=1} \Delta_jc_{ji} + e_i, i = 1,\cdots, N$. The asymptotic distribution of $\hat\mathbf{\Delta}_M - \hat\mathbf{\Delta}_R$ is derived for $p$ fixed and $N \rightarrow \infty,$ under some assumptions on the design matrix, on the error distribution $F$ and on the functions generating the respective estimators. The result has several consequences which have an interest of their own; among others, it is shown that to any $M$-estimator corresponds an $R$-estimator such that the estimators asymptotically equivalent, and conversely. A special case when $\hat\mathbf{\Delta}_M$ is the maximum likelihood estimator and $\hat\mathbf{\Delta}_R$ the $R$-estimator, both asymptotically efficient for some distribution $G$, is also considered.

A strengthened version of the Glivenko-Cantelli theorem for the uniform empirical distribution function is proved. The strengthened Glivenko-Cantelli theorem is used to establish strong laws of large numbers for linear functions of order statistics.

A general law of the iterated logarithm for linear combinations of order statistics is proved. The key tools are (1) iterated logarithm convergence of the uniform empirical process $U_n$ in $\rho_q$-metrics due to B. R. James and (2) almost sure "nearly linear" bounds for the empirical distribution function. A law of the iterated logarithm for the quantile process is also established.

A "moving set inequality," a variant of the one considered by Anderson (1955) and Sherman (1955), is shown to yield a class of random variables whose absolute values are "associated." In particular, a model generated by "contaminated independence" forms the principal example. Further, it is proved that "concordant" functions of associated random variables are associated and then this result is applied to obtain a variety of probability inequalities related to multivariate normal and other distributions. These results generalize the ones obtained by Sidak (1967, 1968, 1971, 1973).

An $n$-variate distribution function is said to be positive dependent by mixture (PDM) if it is a mixture of independent $n$-variate distributions with equal marginals. PDM distributions arise in various contexts of reliability and other areas of statistics. We give a necessary and sufficient condition, by means of independent random variables, for an $n$-variate distribution function to be PDM. The distributions and the expectations of the order statistics of PDM and of independent $n$-variate distributions which have the same marginals, are compared and the results applied to obtain bounds for the reliability of certain "$k$ out of $n$" systems. A characterization of vectors of expectations of order statistics of PDM distribution is shown. Surprisingly many exchangeable distributions are found to be PDM. We prove a closure property of the class of PDM distributions and list some examples.

In the context of the two-action, linear loss, exponential family multiple decision problem, Van Houwelingen (1973), (1976) has shown that faster rates of convergence are deducible for monotone empirical Bayes procedures than result from application of the bound established for general empirical Bayes procedures by Johns and Van Ryzin (1972). This note generalizes a (1973) Van Houwelingen bound to arbitrary $k$-action, monotone loss, MLR family multiple decision problems. An example is given to show that the result is a useful alternative to the recent Van Ryzin and Susarla (1977) multiple decision problem generalization of Johns and Van Ryzin.

Consider a regular $p$-dimensional exponential family such that either the distributions are concentrated on a lattice or they have a component whose $k$-fold convolution has a bounded density with respect to Lebesgue measure. Then, if a parametric function has an unbiased estimator, the minimum variance unbiased estimators are asymptotically equivalent to the maximum likelihood estimators; and, hence, are asymptotically efficient. Examples are given to show that a condition like the above is needed to obtain the asymptotic equivalence.

The rate at which the mean integrated square error decreases as sample size increases is evaluated for general $L^1$ kernel estimates and for the Fourier integral estimate for a probability density. The rates are compared to that of the minimum M.I.S.E.; the Fourier integral estimate is found to be asymptotically optimal.

Let $X_1,\cdots, X_n$ be independent, identically distributed random vectors with values in $\mathbb{R}^d$ and with a common probability density $f$. If $V_k(x)$ is the volume of the smallest sphere centered at $x$ and containing at least $k$ of the $X_1,\cdots, X_n$ then $f_n(x) = k/(nV_k(x))$ is a nearest neighbor density estimate of $f$. We show that if $k = k(n)$ satisfies $k(n)/n \rightarrow 0$ and $k(n)/\log n \rightarrow \infty$ then $\sup_x|f_n(x) - f(x)|\rightarrow 0$ w.p. 1 when $f$ is uniformly continuous on $\mathbb{R}^d$.

An estimator for the spectral density of a stationary time sequence is introduced. The form of the estimator is motivated by the analogy with the summability theory of Fourier series. A theorem relating to rate of consistency is proved. The effect of a mean correction is considered.

Let $X_1, X_2$ be two independent variables with means $\theta_1, \theta_2$. Then two examples are given (one binomial, one normal) in which an estimator, depending only on $X_2$ is admissible for estimating $\theta_1$.

The property of completeness, for the minimal sufficient statistics, is investigated in the context of life testing when the set of observations is censored at a fixed time or at a fixed order statistic. Nonparametric families are shown to retain completeness for the observed order statistics and some implications regarding unbiased estimators and similar tests are presented. Most of the common parametric models fail to possess completeness of minimal sufficient statistics under censored sampling in the one-sample, two-sample and regression situations.

Let $y_t$ be an order $p$ autoregressive process of the form $y_t + \sum^p_{s=1} \beta_s y_{t-s} = u_t$, where the $u_t$'s are i.i.d. variables with a symmetric distribution $F$ such that $E \log^+ |u_t| < \infty$. For the Yule-Walker version $\beta_T^\ast$ of the least-squares estimate of $\beta = (\beta_1,\cdots, \beta_p)$, it is shown that $T^\frac{1}{2}(\beta_T^\ast - \beta)$ is bounded in probability.

The purpose of this paper is to demonstrate the existence via construction of a complete set of mutually orthogonal $F$-squares of order $n = 4t, t$ a positive integer, with two distinct symbols. The proof assumes that all Hadamard matrices of order $4t$ exist; they are known to exist for all $1 \leqq t \leqq 50$ and for 2$^p$. Two methods of construction, that is, Hadamard matrix theory and factorial design theory, are given; the methods are related, but the approaches differ.

Finney has used orthogonal partitions in the context of the search for higher order (coarser) partitions of given Latin squares. Hedayat and Seiden use the term $F$-square to denote higher order partitions that are orthogonal to both rows and columns. This note is a short expository treatment of orthogonal partitions in general and is based on the identification of a partition with a vector subspace of Euclidean $N$-space $R^N$. This identification is not new as it is part of the usual vector space approach to analysis of variance. This approach puts the concept of orthogonal partitions in a simple light unencumbered by the language of design of experiments. Another advantage is that certain published bounds on the maximum number of orthogonal partitions of specified type are immediate from the dimensionality restriction imposed by $R^N$. In addition, some counting problems are identified which are of possible interest to researchers in design of experiments and combinatorics.

It is shown, under conditions which include invariant sequential probability ratio tests, that the stopping time is always expontially bounded when the null or alternative hypothesis holds, except in a trivial instance.

Given the joint distribution of cause of failure and system lifetime of a series system in which components fail simultaneously with probability 0, it is possible (under mild regularity conditions) to assume a model for the system in which component lifetimes are independent.

Let $X$ be a nonnegative random variable with probability distribution function $F$. Suppose $X_{i,n} (i = 1,\cdots, n)$ is the $i$th smallest order statistics in a random sample of size $n$ from $F$. A necessary and sufficient condition for $F$ to be exponential is given which involves the identical distribution of the random variables $X$ and $(n - i) (X_{i+1,n} - X_{i,n})$ for some $i$ and $n$, $(1 \leqq i < n)$.