Exact boundaries in sequential testing for phase-type distributions

Abstract

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.

A series expansion formula of the scale matrix with applications in change-point detection

“…The CUSUM procedure triggers an alarm at the first moment this reflected process up-crosses a fixed threshold. Contrary to the study of the classical SPRT [1], the observations ζ fail to be i.i.d. Thus, novel and more flexible approaches are required to tackle this problem.…”

“…In other words, we need to construct a continuous-time process X so that the post-jump points of its reflected path coincide with the CUSUM statistic for general observations ζ, which are not necessarily F 0 or F 1 -distributed. Contrary to the SPRT case [1], symmetry is lost when considering reflection at a one-sided boundary. For this reason, we require two approaches, depending on the sign of the tilting parameter θ.…”

“…To the best of our knowledge, Albrecher et al [1] is the only existing work that uses the fluctuation theory of MAPs in sequential analysis problems. In [1], they focus on Wald's sequential probability ratio test (SPRT) in classical binary sequential hypothesis testing, where observations ζ are F 0 or F 1 -distributed. Hence, the LLR process L becomes a random walk.…”

“…That is the barrier where the average run length, when ζ is independent and F 0 -distributed, equals a given parameter. In this case, as in [1], the process X becomes an ordinary Sparre-Andersen process, represented as a MAP modulated by a modification of the Markov chain for the PH distribution of ζ. Using the first passage identity of its reflected process given in [27], the moment generating function (and thus the first moment as well) of the average run length can be written in terms of the corresponding scale matrix, for which a new series expansion formula is derived.…”

“…Hence, it is essential to develop a way to compute the scale matrix to carry out the introduced techniques in practice. This is an important component missing in [1], where the expression of the scale matrix was obtained only for the case ζ are Erlang distributed. In this paper, we derive a new series expansion formula for the scale matrix for MAPs with a constant drift and general finite-activity one-sided jumps.…”