Exact boundaries in sequential testing for phase-type distributions

TL;DR

This paper derives exact decision boundaries for sequential testing between phase-type and tilted distributions, linking ruin theory with matrix scale functions to optimize error rates and sample size.

Contribution

It introduces a novel method using matrix-valued scale functions and ruin theory to determine exact boundaries in sequential tests for phase-type distributions.

Findings

Exact decision boundaries for specified error rates

Method applies to distributions with regularly varying tails

Provides mean sample size calculations

Abstract

We consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. In this setting, we derive exact decision boundaries for given Type I and Type II errors by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated to a certain one-sided Markov additive process. By suitable transformations the results also apply to other types of distributions including some distributions with regularly varying tail.