A Modification of the Sequential Probability Ratio Test to Reduce the Sample Size

Abstract

The sequential probability ratio test is constructed as a sequential test of one simple hypothesis against another. In many instances a parametric form is assumed for the density or (discrete) probability function, and the two simple hypotheses are specified by two values of the parameter. The sequential probability ratio test has an optimum property for these two hypotheses, namely, given such a test there is no other test with at least as low probabilities of Type I and Type II errors and with smaller expected sample sizes under either or both of the two hypotheses. Usually, however, one is interested in the performance of the procedure for more values of the parameter than these two. A disadvantage of the sequential probability ratio test is that in general the expected sample size is relatively large for values of the parameter between the two specified ones; that is, in cases in which one does not care greatly which decision is taken, a large number of observations is expected. The question is how to reduce the expected sample size for values of the parameter when this tends to be large. In this paper we consider a special case of the problem, when the distribution is normal with known variance and the parameter of interest is the mean. The sequential probability ratio test in this case consists in taking observations sequentially and after each observation is taken comparing the sum of the observations (referred to a suitable origin) with two constants. In this study the two constants are replaced by two linear functions of the number of observations taken, and the taking of observations is truncated (Section 2). Approximations to the operating characteristic (or power function) and the average sample size number are given (Section 4 and 5). Computations for two cases of special interest show a considerable decrease in average sample size at parameter values between the two specified ones (Section 3). The problem is studied by replacing the sum of observations by the Wiener stochastic process (of a continuous time parameter); this can be thought of intuitively as interpolating between observations in a manner consistent with the addition of independent random variables. For this procedure we calculate exactly the operating characteristic, the distribution of observation time, the expected observation time, and related probabilities.