Performance bounds for robust decision problems with uncertain statistics

This paper is concerned with performance bounds for binary hypothesis testing problems based on I. Independent identically distributed observations with uncertainty in the distribution. II. Discrete-time stationary Gaussian observations with spectral uncertainty. III. Continuous-time Poisson observations with uncertain rate functions. We consider uncertainty classes determined by 2-alternating (Choquet) capacities. A robust decision design based on the likelihood ratio test between the least-favorable pair of probability distributions, spectral measures, or rate functions, respectively, is derived. It is shown that for all elements in the uncertainty class this choice of the likelihood ratio guarantees the exponential convergence of the error probability (upper) bounds to zero as the number of observations (or the length of the observation interval in the continuous-time case) increases. Furthermore, the elements of the least-favorable pair are closer to each other in the sense of (directed or indirected) divergence and Chernoff distance than any other elements in the uncertainty class.