Uniform hypothesis testing for ergodic time series distributions

TL;DR

This paper develops a uniform hypothesis testing framework for stationary ergodic time series, establishing conditions for the existence of consistent tests without assuming independence or mixing, based on topological properties.

Contribution

It introduces topological conditions under which consistent uniform tests can be constructed for ergodic time series distributions, without relying on independence assumptions.

Findings

Identifies necessary and sufficient topological conditions for test existence

Provides a framework for uniform error bounds in ergodic time series testing

Extends hypothesis testing to broader classes of dependent processes

Abstract

Given a discrete-valued sample $X_1,...,X_n$ we wish to decide whether it was generated by a distribution belonging to a family $H_0$, or it was generated by a distribution belonging to a family $H_1$. 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 $\epsilon$ there exist a sample size $n$ such that probability of error is upper-bounded by $\epsilon$ for samples longer than $n$. We find some necessary and some sufficient conditions on $H_0$ and $H_1$ under which a consistent test (with this notion of consistency) exists. These conditions are topological, with respect to the topology of distributional distance.