Testing composite hypotheses about discrete ergodic 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 (in particular, no independence or mixing rate assumptions). We find some necessary and some sufficient conditions, formulated in terms of the topological properties of H0 and H1, for the existence of a consistent test. For the case where H1 is the complement of H0 (to the set of all stationary ergodic processes), these necessary and sufficient conditions coincide, thereby providing a complete characterization of families of processes membership to which can be consistently tested, against their complement, based on sampling. This criterion includes as special cases several known and some new results on testing for membership to various parametric families, as well as testing identity, independence, and other hypotheses.