Sequential Interval Estimation for the Means of Normal Populations

Abstract

The theory of sequential analysis was originally developed by Wald [12] in the context of testing a simple hypothesis against a specific alternative. Wald and Stein [11] subsequently considered the problem of finding a sequential confidence interval of prescribed width $(w)$ and confidence coefficient $(1 - \alpha)$ for the mean $(m)$ of a normal distribution with known variance $(\sigma^2)$. Their basic result, to the effect that no sequential procedure existed for this problem which had an average sample size less than the number of measurements required by the classical single-sample procedure, implies a substantial limitation as to what sequential interval estimation for the mean can accomplish in the case of a normal distribution. However, as we hope to show in this paper, by suitably redefining the problem, we can find sequential procedures either for obtaining a confidence interval for the difference in the means of two normal populations or for obtaining simultaneous confidence intervals for the means of $k$ normal populations, which promise to be useful in some applications. The basic idea in the reformulation of the problem is that the requirement to be put on the width of the confidence interval should depend on the location of the confidence limits, either with respect to some standard value or with respect to the confidence limits for the means of other populations. When this reformulation is appropriate to the problem at hand, Tables I and II indicate that a substantial saving is possible with the sequential procedure developed here at the risk of a relatively small (about 15 or 20%) increase in the average sample size if the least favorable parameter configuration should occur. In the next section we will find a sequence of random intervals $(J_n)$ such that $P\lbrack m \varepsilon J_n$ for all $n, n_0 \leqq n \leqq T\rbrack \geqq 1 - \alpha$. When $T = \infty$, sequences of this type, which might be called "confidence sequences" seem to have been first introduced into statistics by Wald in Chapter 10 of [12]. Such sequences were used by the present writer [7], [8], [9] as an important tool in finding sequential solutions to problems involving the selection of one of a finite number of possible decisions. Recently Darling and Robbins [2], [3], [4], [5] Robbins and Siegmund [10] have introduced and studied the properties of some new types of confidence sequences. Their work will be commented on briefly at the end of the paper.