Constructing Confidence Regions of Optimal Expected Size

Abstract

This article presents a Monte Carlo method for approximating the minimax expected size (MES) confidence set for a parameter known to lie in a compact set. The algorithm is motivated by problems in the physical sciences in which parameters are unknown physical constants related to the distribution of observable phenomena through complex numerical models. The method repeatedly draws parameters at random from the parameter space and simulates data as if each of those values were the true value of the parameter. Each set of simulated data is compared to the observed data using a likelihood ratio test. Inverting the likelihood ratio test minimizes the probability of including false values in the confidence region, which in turn minimizes the expected size of the confidence region. We prove that as the size of the simulations grows, this Monte Carlo confidence set estimator converges to the Γ-minimax procedure, where Γ is a polytope of priors. Fortran-90 implementations of the algorithm for both serial and parallel computers are available. We apply the method to an inference problem in cosmology.