Information Spectrum Approach to Binary Hypothesis Testing with Unknown Parameters*

In the field of information theory, the optimum hypothesis testing exponent, which is defined as the maximum exponent of the type II error probability under the condition that the type I error probability is smaller than or equal to some constant, has been analyzed in several settings. In particular, information spectrum methods, which is one of efficient techniques in information theory, have been applied to the hypothesis testing problem, and have succeeded in giving the general formula of the optimum hypothesis testing exponent. Recently, two typical extensions of the binary hypothesis testing setting have received much attention. One is the case where we do not know the probabilistic distributions. The other is the hypothesis testing problem in the presence of noise. However, information spectrum methods have not yet been applied to the hypothesis testing in these two directions. Hence, in this paper we develop information spectrum methods to treat the hypothesis testing in these settings. In particular, we first consider the hypothesis testing with noise and show the optimum exponent of the type II error probability under the condition that the type I error probability is smaller than or equal to some constant. Then, we extend this result to the case where probability distributions of data are unknown.