Finite Sample-Size Regime of Testing Against Independence with Communication Constraints
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The central problem of Hypothesis Testing (HT) consists in determining the error exponent of the optimal Type II error for a fixed (or decreasing with the sample size) Type I error restriction. This work studies error exponent limits in distributed HT subject to partial communication constraints. We derive general conditions on the Type I error restriction under which the error exponent of the optimal Type II error presents a closed-form characterization for the specific case of testing against independence. By building on concentration inequalities and rate-distortion theory, we first derive the performance limit in terms of the error exponent for a family of decreasing Type I error probabilities. Then, we investigate the non-asymptotic (or finite sample-size) regime for which novel upper and lower bounds are derived to bound the optimal Type II error probability. These results shed light on the velocity at which the error exponents, i.e. the asymptotic limits, are achieved as the samples grows.
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