Design and performance evaluation in Kiefer-Weiss problems when sampling from discrete exponential families

TL;DR

This paper develops algorithms for optimal sequential hypothesis testing in discrete exponential families within the Kiefer-Weiss framework, providing performance evaluation tools and comparing with classical tests.

Contribution

It introduces new algorithms for designing optimal tests in the Kiefer-Weiss problem for discrete exponential families and offers performance evaluation formulas.

Findings

Algorithms implemented in R for binomial, Poisson, and negative binomial distributions.

Performance formulas for operating characteristic, average sample number, and related metrics.

Numerical comparisons showing the Kiefer-Weiss solution's advantages over SPRT and fixed-sample tests.

Abstract

In this paper, 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 the Kiefer-Weiss problems. For the observations which follow a discrete exponential family, we provide algorithms for optimal design in the modified Kiefer-Weiss problem, and obtain formulas for evaluating their performance, calculating operating characteristic function, average sample number, and some related characteristics. These formulas cover, as a particular case, the sequential probability ratio tests (SPRT) and their truncated versions, as well as optimal finite-horizon sequential tests. On the basis of the developed algorithms we propose a method of construction of optimal tests and their performance evaluation for the original Kiefer-Weiss problem. All the algorithms are implemented as functions in the R programming language and can be downloaded from {\tt https://github.com/tosinabase/Kiefer-Weiss}, where the functions for the binomial, Poisson, and negative binomial distributions are readily available. Finally, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test having the same levels of the error probabilities.