# First- and Second-Order Hypothesis Testing for Mixed Memoryless Sources with General Mixture

## TL;DR

This paper derives the optimal exponents for first- and second-order hypothesis testing between mixed memoryless sources, extending to more general sources and relating to compound hypothesis testing.

## Contribution

It provides the first- and second-order epsilon-exponents for hypothesis testing involving mixed memoryless sources, generalizing previous results and connecting to broader hypothesis testing frameworks.

## Key findings

- Derived second-order epsilon-exponent for mixed null and stationary alternative hypotheses.
- Extended results to cases where both hypotheses are mixed memoryless sources.
- Discussed implications for general source and compound hypothesis testing.

## 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 epsilon, is called the epsilon-optimum exponent. In this paper, we first give the second-order epsilon-exponent in the case where the null hypothesis and the 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. We address the first-order epsilon-optimum exponent in this setting. In addition, an extension of our results to more general setting such as the hypothesis testing with mixed general source and the relationship with the general compound hypothesis testing problem are also discussed.