Uniform hypothesis testing for finite-valued stationary processes

Daniil Ryabko 1, *
* Corresponding author
1 SEQUEL - Sequential Learning
LIFL - Laboratoire d'Informatique Fondamentale de Lille, LAGIS - Laboratoire d'Automatique, Génie Informatique et Signal, Inria Lille - Nord Europe
Abstract : Given a discrete-valued sample $X_1,\dots,X_n$ 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.
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Daniil Ryabko. Uniform hypothesis testing for finite-valued stationary processes. Statistics, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2014, 48 (1), pp.121-128. ⟨10.1080/02331888.2012.719511⟩. ⟨inria-00610009v2⟩

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