Minimax Optimal Procedures for Joint Detection and Estimation

TL;DR

This paper develops minimax optimal procedures for joint hypothesis testing and parameter estimation under distributional uncertainties, introducing new algorithms and numerical results for practical implementation.

Contribution

It proposes a unified framework for joint detection and estimation with distributional uncertainty, including modified algorithms for faster convergence and stability.

Findings

Optimal policies induce an $f$-similarity to identify least favorable distributions.

Modified algorithms improve convergence speed and numerical stability.

Numerical results validate the theoretical procedures for band-type uncertainty models.

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.