Uniform hypothesis testing for finite-valued stationary processes

Abstract

Given a discrete-valued sample X₁, …, X_n, 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 Types I and II) is uniformly bounded. More precisely, we require that for each ϵ there exists 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.

Keywords:

• hypothesis testing
• ergodic processes
• stationary processes
• composite hypotheses

Acknowledgements

This work has been partially supported by the French Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the ‘Contrat de Projets Etat Region (CPER) 2007-2013’.