Numerical solution of Kiefer-Weiss problems when sampling from continuous exponential families

Abstract

In this article, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in Kiefer-Weiss problems. The main goal of the Kiefer-Weiss problem is designing hypothesis tests that minimize the maximum average sample number, over all parameter values, as opposed to both the sequential probability tests (SPRTs) minimizing the average sample number only at two hypothesis points and the classical fixed-sample-size test. For observations that follow a distribution from an exponential family of the continuous type, we provide algorithms for optimal design in the modified Kiefer-Weiss problem and obtain formulas for evaluating the performance of sequential tests by calculating the operating characteristic function, the average sample number, and some related characteristics. These formulas cover, as a particular case, the SPRTs and their truncated versions, as well as optimal finite-horizon sequential tests. In the setting of the original Kiefer-Weiss problem we apply the method of our recent work (Sequential Analysis 2022, 41(2), 198–219) for numerical construction of the optimal tests. For the particular case of sampling from a normal distribution with a known variance, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test provided that the three tests have the same levels of the error probabilities. All of the algorithms are implemented in the form of computer code written in the R programming language and are available at the GitHub public repository (https://github.com/tosinabase/Kiefer-Weiss). Guidelines on the adaptation of the program code to other exponential family distributions are provided.

Keywords:

• Exponential family
• hypothesis testing
• Kiefer-Weiss problem
• optimal sequential tests
• optimal stopping
• sequential analysis

AMS CLASSIFICATION:

• 62L10
• 62L15
• 62F03
• 60G40
• 62M02

ACKNOWLEDGMENTS

The authors are grateful to the anonymous reviewers and the Associate Editor for thoroughly reading an earlier version of the article and for their very helpful comments and suggestions.

DISCLOSURE

The authors have no conflicts of interest to report.

Additional information

Funding

The work of A. Novikov (Russia) is partially supported by the Mathematical Center in Akademgorodok under Agreement No. 075-15-2022-281 (05.04.2022) with the Ministry of Science and Higher Education of the Russian Federation. A. Novikov (Mexico) gratefully acknowledges partial support of Sistema Nacional de Investigadores (National Researchers System) by CONACyT (Mexico) for this work. F. Farkhshatov thanks CONACyT (México) for scholarship for his doctoral studies.