A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

In statistical language we seek to give a lower bound on the performance of any estimator over a class of problems (often called the minimax risk over the class). In the language of information theory, we speak of converse results, which give performance bounds for all communication schemes over a noisy channel. In the statistics literature, one standard approach to proving converse results is via Fano's inequality (see [1, Theorem 2.11.1]). However, recent information-theoretic literature has shown how to obtain sharper converse bounds. The resulting improvements can be significant at finite sample size, and give bounds that are close to optimal, as illustrated in the work of Polyanskiy, Poor and Verdú [2]. The present paper shows how the method of [2], although developed for channel coding problems, gives stronger risk lower bounds for high-dimensional estimation problems, compared to the standard Fano approach. We first describe the general setup, following the treatment and notation of [3, Chapter 2].