Achievable Error Exponents for Almost Fixed-Length Hypothesis Testing and Classification

We revisit multiple hypothesis testing and propose a two-phase test, where each phase is a fixed-length test and the second-phase proceeds only if a reject option is decided in the first phase. We derive achievable error exponents of error probabilities under each hypothesis and show that our two-phase test bridges over fixed-length and sequential tests in the similar spirit of Lalitha and Javidi (ISIT, 2016) for binary hypothesis testing. Specifically, our test could achieve the performance close to a sequential test with the asymptotic complexity of a fixed-length test and such test is named the almost fixed-length test. Motivated by practical applications where the generating distribution under each hypothesis is \emph{unknown}, we generalize our results to the statistical classification framework of Gutman (TIT, 1989). We first consider binary classification and then generalize our results to $M$-ary classification. For both cases, we propose a two-phase test, derive achievable error exponents and demonstrate that our two-phase test bridges over fixed-length and sequential tests. In particular, for $M$-ary classification, no final reject option is required to achieve the same exponent as the sequential test of Haghifam, Tan, and Khisti (TIT, 2021). Our results generalize the design and analysis of the almost fixed-length test for binary hypothesis testing to broader and more practical families of $M$-ary hypothesis testing and statistical classification.