A criterion for hypothesis testing for stationary processes

TL;DR

This paper establishes a criterion for hypothesis testing of stationary ergodic processes, providing necessary and sufficient conditions for the existence of consistent tests without assuming specific distributional forms.

Contribution

It introduces a new criterion for the existence of consistent tests for stationary ergodic processes, extending previous results to parametric families and analyzing stronger consistency notions.

Findings

Necessary and sufficient conditions for consistent tests are provided.

The criterion applies to parametric family membership testing.

Conditions for finite-sample guarantees are discussed.

Abstract

Given a finite-valued sample $X_1,...,X_n$ we wish to test whether it was generated by a stationary ergodic process belonging to a family $H_0$, or it was generated by a stationary ergodic process outside $H_0$. We require the Type I error of the test to be uniformly bounded, while the type II error has to be mande not more than a finite number of times with probability 1. For this notion of consistency we provide necessary and sufficient conditions on the family $H_0$ for the existence of a consistent test. This criterion is illustrated with applications to testing for a membership to parametric families, generalizing some existing results. In addition, we analyze a stronger notion of consistency, which requires finite-sample guarantees on error of both types, and provide some necessary and some sufficient conditions for the existence of a consistent test. We emphasize that no assumption on the process distributions are made beyond stationarity and ergodicity.