An Efficient Approximate Solution to the Kiefer-Weiss Problem

The problem is to decide on the basis of repeated independent observations whether θ₀ or θ₁ is the true value of the parameter θ of a Koopman-Darmois family of densities, where the error probabilities are at most α₀ and α₁. 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 α⁻¹₀)1/2) as α₀ and α₁ go to 0, subject to the condition that 0 < C₁ < log α₀/logα₁ < C₂ < ∞ for fixed but arbitrary constants C₁ and C₂. 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.