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A criterion for hypothesis testing for stationary processes

29 May 2009
D. Ryabko
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Abstract

Given a finite-valued sample X1,...,XnX_1,...,X_nX1​,...,Xn​ we wish to test whether it was generated by a stationary ergodic process belonging to a family H0H_0H0​, or it was generated by a stationary ergodic process outside H0H_0H0​. 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 H0H_0H0​ 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.

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