2-SPRT'S and The Modified Kiefer-Weiss Problem of Minimizing an Expected Sample Size

Abstract

A simple combination of one-sided sequential probability ratio tests, called a 2-SPRT, is shown to approximately minimize the expected sample size at a given point $\theta_0$ among all tests with error probabilities controlled at two other points, $\theta_1$ and $\theta_2$. In the symmetric normal and binomial testing problems, this result applies directly to the Kiefer-Weiss problem of minimizing the maximum over $\theta$ of the expected sample size. Extensive computer calculations for the normal case indicate that 2-SPRT's have efficiencies greater than 99% regardless of the size of the error probabilities. Accurate approximations to the error probabilities and expected sample sizes of these tests are given.