An Efficient Approximate Solution to the Kiefer-Weiss Problem

Abstract

The problem is to decide on the basis of repeated independent observations whether $\theta_0$ or $\theta_1$ is the true value of the parameter $\theta$ of a Koopman-Darmois family of densities, where the error probabilities are at most $\alpha_0$ and $\alpha_1$. An explicit method is derived for determining a combination of one-sided SPRT's, known, as a 2-SPRT, which minimizes the maximum expected sample size to within $o((\log \alpha^{-1}_0)^{1/2})$ as $\alpha_0$ and $\alpha_1$ go to 0, subject to the condition that $0 < C_1 < \log \alpha_0/\log\alpha_1 < C_2 < \infty$ for fixed but arbitrary constants $C_1$ and $C_2$. For the case of testing the mean of an exponential density, extensive computer calculations comparing the proposed 2-SPRT with optimal procedures show that the 2-SPRT comes within 2% of minimizing the maximum expected sample size over a broad range of error probability and parameter values.