On the strong consistency, weak limits and practical performance of the ML estimate and Bayesian estimates of a symmetric domain in $R^k$

Abstract

This paper considers a problem of estimating an unknown symmetric region in $R^k$ based on $n$ points randomly drawn from it. The domain of interest is characterized by two parameters: size parameter $r$ and shape parameter $p$. Three methods are investigated which are the maximum likelihood, Bayesian procedures, and a composition of these two. A modification of Wald's theorem as well as a Bayesian version of it are given in this paper to demonstrate the strong consistency of these estimates. We use the measures of symmetric differences and the Hausdorff distance to assess the performance of the estimates. The results reveal that the composite method does the best. Discussion on the convergence in distribution is also given.