Fading Boundaries: On a Nonparametric Variant of the Kiefer--Weiss Problem

A nonparametric variant of the Kiefer--Weiss problem is proposed and investigated. In analogy to the classical Kiefer--Weiss problem, the objective is to minimize the maximum expected sample size of a sequential test. However, instead of taking the maximum over a parametric family of distributions, it is taken over all distributions defined on the given sample space. Two optimality conditions are stated, one necessary and one sufficient. The latter is based on existing results on a more general minimax problem in sequential detection. These results are specialized and made explicit in this paper. It is shown that the nonparametric Kiefer--Weiss test is distinctly different from its parametric counterpart and admits non-standard, arguably counterintuitive properties. In particular, it can be nontruncated and critically depends on its stopping rules being randomized. These properties are illustrated numerically using the example of coin flipping, that is, testing the success probability of a Bernoulli random variable.