For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. Let $t_1, t_2,\cdots$ be a sequence of points in $T$ and let $Y_1, Y_2,\cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and which minimize $\sum^n_{i=1} \lbrack\hat{m}_n(t_i; Y_1,\cdots, Y_n) - Y_i\rbrack^2$ are already known. Brunk has investigated their consistency when $N = 1$. In this paper additional consistency results are obtained when $N = 1$ and some results are obtained in the case $N = 2$. In addition, we prove several lemmas about the law of large numbers which we believe to be of independent interest.

We consider the partial order on the unit square; $s_1 = (x_1, y_1) \ll s_2 = (x_2, y_2)$ if and only if $x_i \leqq y_i$ for $i = 1, 2$, and say that a real-valued function $f$ is isotone if $s_1 \ll s_2$ implies that $f(s_1) \leqq f(s_2)$. Suppose that for each point, $s$, in the unit square we have a distribution with median $m(s)$ and $m(s)$ is isotone. In this paper we propose an isotone estimator for $m$ which we denote by $\hat{m}$ and give an algorithm for computing $\hat{m}$. Furthermore we show that if $x_{ij} (j = 1, \cdots, n_i)$ are observations at $s_i (i = 1, \cdots, k)$ then $\hat{m}$ minimizes $D(f) = \sum^k_{i=1} \sum^{n_i}_{j=1} |f(s_i) - x_{ij}|$ over all isotone functions $f$. The estimator is also shown to be consistent for $m$ and some rates are given for this convergence. A brief discussion of isotone percentile regression is also given.

It is assumed that a random sample of size $n$ is taken from a bivariate distribution whose density $f(x, y)$ possesses a Monotone Likelihood Ratio, i.e. for all $x_1 < x_2$ and $y_1 < y_2, f(x_1, y_1)f(x_2, y_2) \geqq f(x_1, y_2)f(x_2, y_1)$. When the sample is "broken," i.e. when the $x$- and $y$-values are received in random relative order, it is desirable to optimally "reconstruct" the original bivariate sample. Optimal properties of the Maximum Likelihood Pairing (MLP) of $x$- and $y$-values, obtained by DeGroot, et al. in [1], are generalized to the class of distributions defined above, with particular attention given to the trinomial distribution. In addition, one of the main results shown is that in general the MLP is better than random pairing, in that the expected number of correct pairings using the MLP is greater than unity.

Under the assumption of a uniform parent distribution, we find an exact expression for the mean-square error of an order-statistic approximation suggested by R. R. Bahadur. From this result we obtain an asymptotic formula for the mean-square error when the parent distribution is not necessarily uniform by transforming with the inverse of the cumulative distribution function of the parent.

Let $\Theta$ be an open subset of $R^k$ and for each $\theta \in \Theta$, let $X_1, \cdots, X_n$ be independent rv's defined on the probability space $(\mathscr{X}, \mathscr{A}, P_\theta)$, and let $p_{j, \theta}$ be the distribution of the rv $X_j$. Let $f_j(\bullet; \theta)$ be a specified version of the Radon-Nikodym derivative of $p_{j, \theta}$ with respect to a $\sigma$-finite measure $\mu$ and set $f_j(\theta) = f_j(X_j; \theta)$. Furthermore, for $\theta, \theta^\ast \in \Theta$, set $\phi_j(\theta, \theta^\ast) = \lbrack f_j(\theta^\ast)/f_j(\theta)\rbrack^{\frac{1}{2}}$ and suppose that $\phi_j(\theta, \theta^\ast)$ is differentiable in quadratic mean (qm) with respect to $\theta^\ast$ at $(\theta, \theta)$, when the probability measure $P_\theta$ is employed, with qm derivative $\dot{\phi}_j(\theta)$. Set $\Delta_n(\theta) = 2n^{-\frac{1}{2}} \sum^n_{j=1} \dot{\phi}_j(\theta), \Gamma_j(\theta) = 4\mathscr{E}_\theta\lbrack\dot{\phi}_j(\theta)\dot{\phi}_j'(\theta)\rbrack, \bar{\Gamma}_n(\theta) = n^{-1} \sum^n_{j=1} \Gamma_j(\theta)$, and suppose that $\bar{\Gamma}_n(\theta) \rightarrow \bar{\Gamma}(\theta)$ and $\bar{\Gamma}(\theta)$ is positive definite on $\Theta$. Finally, for $h_n \rightarrow h \in R^k$, set $\theta_n = \theta + h_nn^{-\frac{1}{2}}$ and $\Lambda_n(\theta) = \log\lbrack dP_{n,\theta_n}/dP_{n,\theta}\rbrack$, where $P_{n,\theta}$ stands for the restriction of $P_\theta$ to $\mathscr{A}_n = \sigma(X_1, \cdots, X_n)$. Then, under suitable--and not too hard to verify--conditions, we obtain, the following results. The limits are taken as $n \rightarrow \infty$. THEOREM 1. $\Lambda_n(\theta) - h'\Delta_n(\theta) \rightarrow -\frac{1}{2} h'\bar{\Gamma}(\theta)h$ in $P_\theta$-probability, $\theta \in \Theta$. THEOREM 2. $\mathscr{L}\lbrack\Delta_n(\theta) \mid P_\theta\rbrack \Rightarrow N(0, \bar{\Gamma}(\theta)), \theta \in \Theta$. THEOREM 3. $\mathscr{L}\lbrack\Lambda_n(\theta) \mid P_\theta\rbrack \Rightarrow N(-\frac{1}{2} h'\bar{\Gamma}(\theta)h, h'\bar{\Gamma}(\theta)h), \theta \in \Theta$. THEOREM 4. $\Lambda_n(\theta) - h'\Delta_n(\theta) \rightarrow -\frac{1}{2}h'\bar{\Gamma}(\theta)h$ in $P_{\theta_n}$-probability, $\theta \in \Theta$. THEOREM 5. $\mathscr{L}\lbrack\Lambda_n(\theta) \mid P_{\theta_n}\rbrack \Rightarrow N(\frac{1}{2}h'\bar{\Gamma}(\theta)h, h'\bar{\Gamma}(theta)h), \theta \in \Theta$. THEOREM 6. $\mathscr{L}\lbrack\Delta_n(\theta) \mid P_{\theta_n}\rbrack \Rightarrow N(\bar{\Gamma}(\theta)h, \bar{\Gamma}(\theta)), \theta \in \Theta$.

In this paper it is shown that in order for the trimmed mean to be asymptotically normal, it is necessary and sufficient that the sample be trimmed at sample percentiles such that the corresponding population percentiles are uniquely defined. (The sufficiency of this condition is well known.) In addition, the (non-normal) limiting distribution of the trimmed mean when this condition is not satisfied is derived, and it is shown that in some situations the use of the trimmed mean may lead to severely biased inferences. Some possible remedies are briefly discussed, including the use of "smoothly" trimmed means.

In a previous paper, the authors introduced a new criterion of expectation consistency between probability distributions of discrete data given discrete parameter values and arbitrary posterior probability distributions for the parameter. It is here shown, under very weak assumptions, that expectation consistency implies that the posterior distributions are generalized Bayes. However when the posterior distributions are generalized Bayes, the implied prior distribution need not be unique. The class of implied distributions is characterized in terms of a partition of parameter space.

Anbar (1971) and, independently, Abdelhamid (1971) have shown that if the density $g$ of the errors of estimates of function values is known, a transformation of observations leads to stochastic approximation methods which under mild conditions produce asymptotically efficient estimators (the first author considers the RM case, the second the RM and KW cases). This paper treats the RM case and shows that the same asymptotic result can be achieved without the knowledge of the density $g$.

Standard finite state and action discrete time Markov decision processes with discounting are studied using a new optimality criterion called moment optimality. A policy is moment optimal if it lexicographically maximizes the sequence of signed moments of total discounted return with a positive (negative) sign if the moment is odd (even). This criterion is equivalent to being a little risk adverse. It is shown that a stationary policy is moment optimal by examining the negative of the Laplace transform of the total return random variable. An algorithm to construct all stationary moment optimal policies is developed. The algorithm is shown to be finite.

A Bayesian formulation of the problem of analysing data from a curvilinear regression of $y$ on $x_1, x_2, \cdots, x_r$ in order to predict a future value of $y$ is considered. The problem is to obtain a criterion to decide which is the best subset of $x_1, x_2, \cdots, x_r$ to perform this prediction. Under very strict assumptions the criterion obtained is shown to use the same statistic as the orthodox (least squares) approach.

Let the $p$-component vector $X$ be normally distributed with mean $\xi$ and covariance $\sigma^2I$ where $I$ denotes the identity matrix and $\sigma$ is known. For estimating $\xi$ with quadratic loss, it is known that $X$ is minimax but inadmissible for $p \geqq 3$. We obtain a family of estimators which dominate $X$ and are admissible. These estimators are, therefore, both minimax and admissible.

The present paper deals with the asymptotic theory of sequential confidence intervals of prescribed width $2d (d > 0)$ and prescribed coverage probability $1 - \alpha (0 < \alpha < 1)$ for the (unknown, per unit) strength of bundle of parallel filaments. In this context, certain useful convergence results on the empirical distribution and on the bundle strength of filaments are established and incorporated in the proofs of the main theorems. The results are the sequential counterparts of some fixed sample size results derived in a concurrent paper of Sen, Bhattacharyya and Suh [9].

A rigorous proof is given of the often stated fact that if the variance of an unbiased estimator of a function of a real parameter attains the Cramer-Rao lower bound then the family of distributions must be a one-parameter exponential family.

Let $\mathscr{P}$ be a family of probability measures defined on a $\sigma$-field $\mathscr{A}$ on $X$ and $G$ be a group of transformations on $X$ such that $Pg^{-1} \in \mathscr{P}$ for all $P \in \mathscr{P}, g \in G$. Let $\mathscr{A}_I$ be the $\sigma$-field of $G$-invariant sets of $\mathscr{A}$ and $\mathscr{A}_{I^\ast}$ the $\sigma$-field of $\mathscr{P}$-almost $G$-invariant sets of $\mathscr{A}$. Let $\mathscr{A}_S$ be a sufficient $\sigma$-field for $\mathscr{P} \mid \mathscr{A}$. Hall, Wijsman and Ghosh proved that $\mathscr{A}_S \cap \mathscr{A}_I$ is sufficient for $\mathscr{P} \mid \mathscr{A}_I$ if $g\mathscr{A}_S = \mathscr{A}_S$ for each $g \in G$ and $\mathscr{A}_S \cap \mathscr{A}_I \sim \mathscr{A}_S \cap \mathscr{A}_{I^\ast}(\mathscr{P})$. They posed the question whether the first condition alone suffices to prove this result. An example shows that the answer is no. For dominated families we show that $\mathscr{A}_S \cap \mathscr{A}_{I^\ast}$ is always sufficient for $\mathscr{P} \mid \mathscr{A}_{I^\ast}$, a result which is not true any more for undominated families.

Sufficient conditions for admissibility of the best invariant confidence interval for a translation or scale parameter are given, for a very wide class of loss functions. These conditions result by adapting a theorem of L. D. Brown [2]. Simpler sufficient conditions are found for a subclass of loss functions of special interest. The subclass of losses involves three components. One concerned with coverage of the true value, another concerned with the distance from the interval end points to the true parameter, and a third concerned with length of the interval. Such a loss function unifies confidence interval and point estimation in the sense that if an optimality property holds for all loss functions in the subclass, then the optimality property holds for typical confidence interval problems and typical point estimation problems.

The construction of proper Bayes tests which are invariant under non-compact groups has been treated in Kiefer and Schwartz (1965) and Schwartz (1966, 1969). These papers are limited to testing problems concerning exponential families' $\operatorname{pdf}$'s. The present note shows, by means of two simple examples, that the same methods can sometimes be applied more generally. One example considers estimation of the ratio of the mean to the variance for the normal distribution. The second example considers testing problems in which the pdf's have an exponential factor as well as a non-exponential factor.

The purpose of this note is to show that asymptotically distribution-free (ADF) rank tests proposed by Koul [Ann. Math. Statist. 41 (1970), 1273-1281] for testing subhypotheses in multiple linear regression when based on symmetric scores are asymptotically totally inefficient and can even be inconsistent for symmetric parent distributions. Also, the condition given in his Theorem 2.4 for a test to be ADF is not necessary.

Suppose $g$ is a probability density on $R = (-\infty, +\infty)$ with a continuous derivative and with $I(g) = \int (g'/g)^2g < + \infty$. Suppose $\{\varepsilon_n\}$ is a sequence of positive numbers converging to zero and $V_i, Z_i$ are random variables, $Z_i \rightarrow 0$ with probability one and $V_n$ is conditionally (given $V_1, \cdots, V_{n-1}, Z_1, \cdots, Z_n$) distributed according to $g$. Estimates $h_n$ of $g'/g$ are constructed, which are based on $V_1 + Z_1, \cdots, V_n + Z_n$ and have the following properties. For almost all $\omega, h_n(\omega, \bullet) \rightarrow g'/g$ on $\{t; g(t) > 0\}$. If $n\varepsilon_n \uparrow + \infty, \sum n^{-1}\varepsilon_n^{-1}|Z_n| < + \infty$ a.e. then for almost all $\omega, h_n(\omega, \bullet) \rightarrow g'/g$ in $L_2(g)$ and $\int h_n^2 dG_n \rightarrow I(g)$ where $G_n$ is the empirical distribution function of $V_1 + Z_1, \cdots, V_n + Z_n$. The results on the pointwise and $L_2(g)$ convergences hold also if $h_n$ are replaced by $h_n(\omega, \nu + \eta_n(\nu, \omega))$ provided $\eta_n$ are small and preserve the measurability of the estimators.

The problems of asymptotically optimum spacing of sample quantiles in connection with asymptotically best linear unbiased estimation of the location parameter of an extreme value distribution of Type I or the scale parameter of an extreme value distribution of Type II or III, are essentially the same. Previous results for the scale parameter of an exponential distribution can therefore be used.

For one sample rank order statistics, a law of iterated logarithm and almost sure convergence to Wiener processes are established here. For the one-sample location problem, a sequential test procedure based on rank order statistics is proposed, and with the aid of the earlier results, it is shown that this has power one and arbitrarily small type I error.

Godambe and Joshi (1965) have shown that for any design the Horvitz-Thompson estimator is admissible in the class of all unbiased estimators of a finite population total. In this note we give a simple alternative proof of this result using induction on the size of the population.

Some conjectures made by De Waal [Ann. Math. Statist. 43 (1972) 344-347] on the expectation of elementary symmetric functions of the roots of a noncentral Wishart matrix are proved true. The method of proof is through a simple, though perhaps obscure, property of these elementary symmetric functions and simple properties of the Wishart distribution.

Let $X = (X_1, \cdots, X_k)$ be a random vector with all $X_i \geqq 0$ and $\sum X_i \leqq 1$. Let $k \geqq 2$, and suppose that none of the $X_i$, nor $1 - \sum X_i$ vanishes almost surely. Without any further regularity assumptions, each of two conditions is shown to be necessary and sufficient for $X$ to be distributed according to a Dirichlet distribution or a limit of such distributions. Either condition requires that certain proportions between components of $X$ be independent of one or more other components of $X$.

In this paper a multivariate analogue of the logistic distribution is considered. We suggest two families of multivariate logistic distributions with the property that marginal distributions are of univariate form and discuss some distributional properties of the multivariate distributions.

For the notation and terminology used in this paper we refer the reader to Raghavarao (1971). The powerful method of constructing designs by the method differences is not so far used for $L_2(s)$ designs. The purpose of this note is to develop this method.