Combinations of one-sided sequential probability ratio tests (SPRT's) are shown to be "nearly optimal" for problems involving a finite number of possible underlying distributions. Subject to error probability constraints, expected sample sizes (or weighted averages of them) are minimized to within $o(1)$ asymptotically. For sequential decision problems, simple explicit procedures are proposed which "do exactly what a Bayes solution would do" with probability approaching one as the cost per observation, $c$, goes to zero. Exact computations for a binomial testing problem show that efficiencies of about 97${\tt\%}$ are obtained in some "small-sample" cases.

To analyze the effect of correlation in random samples on the performance of estimators of location, small correlation approximations for the asymptotic variance are found. Approximately optimal estimators (in the asymptotic minimax sense of Huber) are presented and compared to other estimators in terms of maximum asymptotic variance over the class of $\varepsilon$-contaminated normals. The presence of relatively small correlation can drastically inflate variances, and the optimal rules given here offer substantial improvements over previously considered estimators.

It is shown, using a theorem of Choquet, that any separable experiment is a mixture of experiments admitting boundedly complete and sufficient statistics. The experiments possessing these properties are precisely the experiments which are extremal with respect to mixtures. Dominated models for independent observations $X_1, \cdots, X_n$ admitting boundedly (or $L_p$) complete and sufficient statistics, are considered. It is shown that a subset--say $X_1, \cdots, X_m$ where $m < n$--has the same property provided a regularity condition is satisfied. This condition is automatically satisfied when the observations are identically distributed. In the bounded complete case the proof uses the fact that products of experiments are distributive w.r.t. mixtures. More involved arguments are needed for $L_p$ completeness.

Confidence sets in the form of balls of fixed or scaled radius with respect to an arbitrary pseudometric are considered. Easily computable lower bounds for the radius of these confidence sets are derived. As by-products, bounds for the minimax risk are given and a method of deriving multiple decision procedures from point estimators is obtained.

I. J. Good's 1965 conjecture of the unimodality of the likelihood function of a symmetrical compound multinomial distribution is proved by the variation-diminishing property of the Laplace transform. The result is a special case of a several sample version with asymmetrical compounding Dirichlet distributions. The technique of proof is applied to yield similar results for the negative binomial distribution and a two point mixture of Poissons.

Certain probability properties of $c_n(t)$, the empirical characteristic function $(\operatorname{ecf})$ are investigated. More specifically it is shown under some general restrictions that $c_n(t)$ converges uniformly almost surely to the population characteristic function $c(t).$ The weak convergence of $n^{\frac{1}{2}}(c_n(t) - c(t))$ to a Gaussian complex process is proved. It is suggested that the ecf may be a useful tool in numerous statistical problems. Application of these ideas is illustrated with reference to testing for symmetry about the origin: the statistic $\int\lbrack\mathbf{Im} c_n(t)\rbrack^2 dG(t)$ is proposed and its asymptotic distribution evaluated.

Let $u, u_1, u_2, \cdots$ be a sequence of i.i.d. random $k$-vectors and $a_1, a_2, \cdots$ be a sequence of $k$-vectors. Let $S_n = \sum^n_1 a_i'u_i$. For any positive $L$, let $N = \min \{n \geqq 1: |S_n| \geqq L\}$. In case $k = 1$ and all $a_n$'s are equal and nonzero, Stein [5] showed that $N$ is exponentially bounded provided that $u$ is nondegenerate at 0. In this paper, conditions on the $a_n$'s and on $u$ which guarantee the exponential boundedness of $N$ defined above are obtained. The exponential boundedness of $N' = \min \{n \geqq 1: |S_n + C_n| \geqq L\}$, where $C_1, C_2, \cdots$ is an arbitrary sequence of real numbers, is also considered. Some applications are given.

Let $X_{ni}, i = 1, \cdots, n$ be i.i.d. random variables on an arbitrary measurable space $(\mathscr{X}, B)$. Suppose $\mathscr{L}(X_{ni}) = Q_{n1}, i = 1, \cdots, n$ and let $P_0$ be a fixed probability measure on $(\mathscr{X}, B)$. We consider limiting distribution theory for $U$-statistics $T_n = n^{-1} \sum_{i \neq j} Q(X_{ni}, X_{nj})$ (1) under conditions which imply the product measures $Q_n = Q_{n1} \times \cdots \times Q_{n1}, n$ times, are contiguous to the product measures $P_n = P_0 \times \cdots \times P_0, n$ times, and (2) for kernels $Q$ which are symmetric, square-integrable $(\int Q^2(\bullet, \bullet) dP_0 \times P_0 < \infty)$ and degenerate in a certain sense $(\int Q(\bullet, t)P_0(dt) = 0 \mathrm{a.e.} (P_0))$. Applications to chi-square and Cramer-von Mises tests for a simple hypothesis and Cramer-von Mises tests for the case when parameters have to be estimated, are given. A tail sensitive test for normality is introduced.

Assume a searcher is hunting for an object which has been hidden in one of $N$ regions or cells, with initial prior probability $p_i^1$ that it is in cell $i$. Suppose that to each $i$ there corresponds a sequence $\{\alpha_{ij}\}_{j \geqq 1}$ of random variables, where $\alpha_{ij}$ describes the chances that the searcher will fail to find the object on the $j$th search of $i$, given that the object is in $i$. The joint distribution of $\{\alpha_{ij}: 1 \leqq i \leqq N, j \geqq 1\}$ is known to the searcher. Under a certain monotonicity condition on the $\alpha_{ij}$'s, it is shown that to maximize the probability of finding the object in at most $n_0$ stages of search, the one-stage look ahead rule is optimal. In an earlier paper concerning a related problem, Hall assumed $\{\alpha_{1j}\}_{j \geqq 1}, \cdots, \{\alpha_{N j}\}_{j \geqq 1}$ were independent processes, whereas we allow them to be dependent. Our result is new for independent processes as well.

Lower bounds on the radius of various confidence sets for density functions and their derivatives are determined. In each case investigated, the smallest radius obtainable and the radius actually obtained by using known estimates exhibit the same dependency on the fixed sample size $n$. The lower bounds are derived using methods in Meyer (1976). Although only a few combinations of pseudometric and density class are considered here, the techniques illustrated can be used elsewhere with little conceptual difficulty.

Let $X_1, X_2,\cdots$ be $R^p$-valued random variables having unknown density function $f$. If $K$ is a density on the unit sphere in $R^p, \{k(n)\}$ a sequence of positive integers such that $k(n) \rightarrow \infty$ and $k(n) = o(n)$, and $R(k, z)$ is the distance from a point $z$ to the $k(n)$th nearest of $X_1,\cdots, X_n$, then $f_n(z) = (nR(k, z)^p)^{-1} \sum K((z - X_i)/R(k, z))$ is a nearest neighbor estimator of $f(z).$ When $K$ is the uniform kernel, $f_n$ is an estimator proposed by Loftsgaarden and Quesenberry. The estimator $f_n$ is analogous to the well-known class of Parzen-Rosenblatt bandwidth estimators of $f(z)$. It is shown that, roughly stated, any consistency theorem true for the bandwidth estimator using kernel $K$ and also true for the uniform kernel bandwidth estimator remains true for $f_n$. In this manner results on weak and strong consistency, pointwise and uniform, are obtained for nearest neighbor density function estimators.

It is shown by Takeuchi and Akahira, 1974, that conditional independence together with a condition of "partial sufficiency" imply "prediction sufficiency" for loss functions not depending on the unknown parameter. We shall here prove that these conditions are necessary as well and thereby obtain a complete description, in terms of conditional expectations, of "prediction sufficiency" for loss functions not depending on the unknown parameter. It turns out that these conditions may be replaced by a condition of conditional independence for prior distributions.

For the problem of empirical Bayes classification into two known probability distributions on a finite outcome space, an essentially complete class of procedures is determined. This class is proven to be minimal essentially complete if there are only two possible outcomes.

In the empirical Bayes approach to multiple decision problems, we obtain theorems and lemmas which can be used to obtain asymptotic optimality and rate results in any multiple decision empirical Bayes problem. Applications of these results to a classification problem, a monotone multiple decision, and a selection problem are given. In addition, a special lemma unique to the monotone multiple decision problem gives improved (exact) rate results in that case.

The well-known principle that a decision maker's subjective probabilities are determined from his preferences among finite compound lotteries is developed in an especially transparent approach. This development can be easily modified to give versions of the result in a variety of infinite contexts.

The problem of classifying each of $n$ observations to one of two sub-populations is considered. The classification rule examined chooses that classification with maximum posterior probability. Limiting behavior of the rule is given and several examples are presented which show that the rule can lead to classifying all observations to the same subpopulation. Three simulation studies are reported to indicate that this extreme behavior may occur in small samples.

Consider the problem of estimating $x$ under the inverse linear regression model $Y_i = \alpha + \beta x_i + \varepsilon_i,\quad Z_j = \alpha + \beta x + \varepsilon_j'$ for $i = 1,\cdots, n,\cdots, j = 1,\cdots, m,\cdots,$ where $\{\varepsilon_i\}, \{\varepsilon_j'\}$ are two sequences of i.i.d. rv's with 0 means and finite variances, $\{x_i\}$ is a sequence of known constants and $\alpha, \beta, x$ are unknown parameters. For fixed $T = m + n$, this paper considers a sequential procedure for the optimal allocation of $m$ and $n$. It is shown that, as $T \rightarrow \infty$, the procedure is asymptotically optimal.

Sufficient conditions for the Cramer-Rao inequality are formulated which do not impose any requirement on the estimator and which are independent of the choice of the densities. A hereditary property is proved for these conditions and the attainability of the lower bound is studied.

Let $R^n$ be Euclidean $n$-space and let $O(n)$ be the group of $n \times n$ orthogonal matrices. Consider $\mathscr{F}_0 = \{f\mid f$ is a density on $R^n, f(x) = f(gx), x \in R^n, g \in O(n)\}$, and let $Q = \{q\mid q: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty), q$ is nonincreasing, $\int_{R^n} q(\| x \|^2) dx = 1\}$. If $\Sigma$ is an $n \times n$ positive definite matrix, set $\mathscr{F}_1(\Sigma) = \{f|f(x) = \mid \Sigma|^{-\frac{1}{2}}q(x'\Sigma^{-1}x), q \in Q\}$. For $\mu \in R^1$ and $a_0 \in R^n, \| a_0 \| = 1$, let $\mathscr{F}_2(\mu) = \{f\mid f(x) = q(\| x - \mu a_0\|^2), q \in Q\}$ and $\mathscr{F}_3(\mu) = \{f\mid f(x) = q(\| x - \mu a_0\|^2), q \in Q,\text{and} q \text{convex}\}.$ Uniformly most powerful tests are derived for testing $\mathscr{F}_0$ versus $\mathscr{F}_1(\Sigma)$ and for testing $\mathscr{F}_0$ versus $\{\mathscr{F}_2(\mu) \mid \mu > 0\}$. A uniformly most powerful unbiased test is derived for testing $\mathscr{F}_0$ versus $\{\mathscr{F}_3(\mu) \mid \mu \neq 0\}$.

Let $(X_i, Y_i)$ be $n$ independent rv's having a common bivariate distribution. When the $X_i$ are arranged in nondecreasing order as the order statistics $X_{r:n} (r = 1,2,\cdots, n)$, the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistic. The small-sample theory of the distribution and expected value of the rank $R_{r:n}$ of $Y_{\lbrack r:n\rbrack}$ is studied. In the special case of bivariate normality an illustrative table of the probability distribution of $R_{r,n}$ is given. A more extensive table of $E(R_{r,n})$ is also provided and it is found that asymptotic results require comparatively small finite-sample corrections even for modest values of $n$. Some applications are briefly indicated.

A new formula for cumulants was given by Good (1975). A short proof is now given and the result is used to obtain a new formula for $k$-statistics. This formula can be used both for deriving the expressions of $k$-statistics in terms of power sums of the observations, and for checking and locating errors in formulae that are already in the literature.

A Robbins-Monro type procedure for estimating the zero of a regression function is discussed. The procedure is a modification of the Robbins-Monro procedure which is designed to approximate the zero from below. An almost sure convergence is proved and it is shown that one can guarantee that the procedure overestimate the zero only finitely many times with probability one.

Two inverse sampling procedures, one that uses the classical vector-at-a-time observation rule and another that uses the play-the-winner observation rule, are shown to select the best of $k$ binomial populations with the same probability, independent of the probabilities of success. This shows that the play-the-winner rule is better from the point of view that both the sample size and number of failures of each population are stochastically smaller using play-the-winner than vector-at-a-time.