Abstract: Interval estimation of the largest mean of $k$ normal populations $(k \geqq 1)$ with a common variance $\sigma^2$ is considered. When $\sigma^2$ is known the optimal fixed-width interval is given so that, to have the probability of coverage uniformly lower bounded by $\gamma$ (preassigned), the sample size needed is minimized. This optimal interval is unsymmetric for $k > 2$. When $\sigma^2$ is unknown a sequential procedure is proposed and its behavior is studied. It is shown that the confidence interval obtained, which is also unsymmetric for $k > 2$, behaves asymptotically as well as the optimal interval. This represents an improvement of the procedure of symmetric intervals considered by the author previously; the improvement is significant, especially when $k$ is large.

Abstract: In a previous paper [5], the author has proposed a class of asymptotically optimal (in the sense of Wald [11]) nonparametric tests for testing the hypothesis of no regression in a multiple linear regression model. In the present paper, we are interested in testing that the intercept in the multiple (linear) regression model is zero along with the absence of regression. A class of permutationally distribution-free tests has been proposed and their asymptotic optimality has been established. These results generalize analogus findings of Puri and Sen [9] for ungrouped data. As an important application, an alternative test to the problem of paired comparison has been considered.

Abstract: The purpose of this paper is to find asymptotically optimal Bayes sequential procedures for estimating a function $g(\theta_1, \theta_2,\cdots, \theta_k)$ when there are $k$ experiments $E_1, E_2,\cdots, E_k$ and the performance of the experiment $E_i$ conducts to the observation of a random variable whose distribution depends on the vector parameter $\theta_i$. The term asymptotical refers here to the cost of experimentation tending to zero. The methods used are a generalization of those introduced by Bickel and Yahav.

Abstract: In recent years considerable interests have been given to the pattern classification problem. This problem includes three main aspects, the engineering aspect, the artificial intelligence aspect and the analytical aspect (c. f. [4]). The analytical one is concerned with the mathematical techniques of decision, estimation and optimization under the uncertainty of information. In this paper, our interest will be concentrated on the analytical aspect, especially on the estimation of a discriminant function which is optimum in the sense of the Bayes rule, minize the probability of misclassification, but which is unknown to us. We shall call such the discriminant function an " optimal discriminant function " (o. d. 1.). In previous works (e. g. [4], [7], [8], [9] and [11]), under the given situation of a "training sequence " of observed patterns corectly classified by an external indicator, authors tried to obtain algorithms for finding the o. d. f. on the basis of the training sequence. In general, this approach has been called " learning with a teacher" in the pattern classification problem. And the method of approach which we shall appeal to in this paper is this case. The problem which will be treated in this paper is to classify the observed patterns into two categories, and the procedure which will be used is an application of the " stochastic approximation method " introduced by H. Robbins and S. Monro. Our object is to construct, on the basis of the given training sequence, estimates of the o. d. f. which are asymptotically optimal in the sense that the probabilities of misclassification from the estimates converge (with probability one or in the mean) to the probability of misclassification from the o. d. f. if it is known to us. This paper consists of five sections. In Section 2, we shall give definitions of the optimal discriminant function and of asymptotically optimal estimates to the o. d. f., and we shall prepare several lemmas to be used throughout subsequent sections. In Section 3, we shall treat the case when the o. d. f. is assumed to belong in the L2 space, and give an algorithm for constructing the asymptotically optimal estimates. J. V. Ryzin [7] treated this case, but his algorithm was not recursive and his convergence of the asymptotically optimal estimates was the one in the

Abstract: By examining a particular hypothesis testing problem under a finite memory constraint we derive general guidelines for the design of asymptotically optimal, low complexity, finite memory decision rules. By asymptotically optimal we mean that only a fixed number of bits need be added to memory to achieve the optimal error probability. Thus the fraction of bits "lost" by these low complexity rules tends to zero as memory size becomes large. The rules developed are similar to quantized sequential probability ratio tests.

Abstract: A new sequential design decision rule is proposed for a statistical multiple decision problem with finite state space and with a finite set of available experiments. Conditions are established under which the proposed rule is asymptotically optimal as c, the cost of a single experiment, tends to zero. The rule is compared to those of Chernoff [6] and Box and Hill [5]. In numerical simulations of a type of drug screening experiment, the proposed procedure yielded estimated risks no larger than, often significantly smaller than, those of procedures [6] and [5].