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 ϵ, is called the ϵ-optimum exponent. In this paper, we first give the second-order ϵ-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 ϵ-optimum exponent in this setting.