Uniform hypothesis testing for finite-valued stationary processes

Abstract

Given a discrete-valued sample X1, . . . , Xn we wish to decide whether it was generated by a distribution belonging to a family H0, or it was generated by a distribution belonging to a family H1. In this work we assume that all distributions are stationary ergodic, and do not make any further assumptions (e.g. no independence or mixing rate assumptions). We would like to have a test whose probability of error (both Type I and Type II) is uniformly bounded. More precisely, we require that for each ε there exist a sample size n such that probability of error is upper-bounded by ε for samples longer than n. We find some necessary and some sufficient conditions on H0 and H1 under which a consistent test (with this notion of consistency) exists. These conditions are topological, with respect to the topology of distributional distance.