Minimax Optimal Procedures for Joint Detection and Estimation

Abstract

We investigate the problem of jointly testing a pair of composite hypotheses and, depending on the test result, estimating a random parameter under distributional uncertainties. Specifically, it is assumed that the distribution of the data given the parameter of interest, is subject to uncertainty. Both, a Bayesian formulation and a Neyman-Pearson-like formulation, are considered. It is shown that the optimal policy induces an $f$-similarity that must be maximized to identify the least favorable distributions. Besides the general results, the implementation is investigated using a band-type uncertainty model. For designing the minimax procedures, existing algorithms are modified to increase convergence speed while maintaining numerical stability. The proposed theory is supplemented by numerical results for both formulations.