Efficient Approximate Solutions to the Kiefer-Weiss Problem

Author: Huffman, Michael David

Year: 1980

Degree: Dissertation (Ph.D.)

Advisor: Lorden, Gary A.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/5nn5-n386

Abstract

The problem is to decide on the basis of repeated independent observations whether θ0 or θ1 is the true value of the parameter e of a Koopman-Darmois family of densities, where θ less than θ less than θ. The probability of falsely rejecting e0 is to be at most o:0, and that of falsely rejecting θ1, at most α1. Procedures are studied from the point of view of minimizing the maximum (over θ) expected number of observations required when e is the true value of the parameter.

Two types of tests are considered. The first, based on the well-known sequential probability ratio test (SPRT), dictates after each observation whether to stop and ma.ke a decision, or whether to continue sampling. An explicit method is derived for determining a combination of one-sided SPRT's, known as a 2-SPRT, which minimizes the maximum expected number of observations to within o((n(α0,α1))1/2) as α0 and α1 go to o, where n(α0,α1) is the minimum of the maximum expected sample size, taken over all procedures with error probabilities at most α0 and α1. The second test uses several stages of observations, deciding whether to stop or continue only at the end of each stage. A procedure designed to "do what a sequential test would do", while using at most three stages, is defined and shown to minimize the maximum expected number of observations to within O((n(α0,α1))1/4(log n(α0,α1))3/2) as α0 and α1 go to 0.

Finally, using backward induction, optimal procedures were developed on the computer for the case where the mean of an exponential density is tested. Then extensive computer calculations comparing the proposed 2-SPRT with these optimal procedures show that the 2-SPRT comes within 1% of minimizing the maximum expected sample size over a broad range of error probability and parameter values.

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