Abstract
The first- and second-order optimum achievable exponents in the simple hypothesis testing problem are investigated. The optimum achievable exponent for type II error probability, under the constraint that the type I error probability is allowed asymptotically up to , is called the -optimum exponent. In this paper, we first give the second-order -optimum exponent in the case where the null hypothesis and alternative hypothesis are a mixed memoryless source and a stationary memoryless source, respectively. We next generalize this setting to the case where the alternative hypothesis is also a mixed memoryless source. Secondly, we address the first-order -optimum exponent in this setting. In addition, an extension of our results to the more general setting such as hypothesis testing with mixed general source and a relationship with the general compound hypothesis testing problem are also discussed.
Highlights
National Institute of Information and Communications Technology (NICT), Tokyo 184-8795, Japan; School of Network and Information, Senshu University, Kanagawa 214-8580, Japan
For the case of mixed general sources with finite discrete mixture, we reveal the deep relationship with the compound hypothesis testing problem
The general limiting formula for Bε ( R|X||X) is given as follows, which is the second-order counterpart of Theorem 1, and is utilized to establish Theorem 4 3.2 to give a second-order single-letter formula for hypothesis testing in the case where the null hypothesis is mixed memoryless and the alternative hypothesis is stationary memoryless
Summary
The result is a substantial generalization of that of Strassen [4] We generalize this setting to the case where both null and alternative hypotheses are mixed memoryless X, X to establish the single-letter first-order ε-optimum exponent. Theorem 5 to establish a first-order single-letter formula for hypothesis testing in the case where both the null and alternative hypotheses are mixed memoryless. We give the general formula (Theorem 2) for the second-order ε-optimum exponent, which is used to prove Theorem 4 to establish a second-order single-letter formula for hypothesis testing in the case where the null hypothesis is mixed memoryless and the alternative hypothesis is stationary memoryless.
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