Inference about an Unknown Compact Domain in R, k

Abstract

We consider a problem with many types of applications originally suggested by David Kendall. The problem is to estimate an unknown compact domain S in Rk when uniformly distributed observations x1,x2,...,xn are available from S. We will report a collection of results on parametric MLEs and Bayes estimates of the set S. The results include both finite sample illustrations and asymptotics. We address the case when the contour of S is not too jagged : the parametric model is that of Lp balls. Assessment of our estimate is achieved by considering the Hausdorff distance between S and its estimate, as well as the volume of the symmetric difference. In particular, the asymptotic distribution of these two quantities is derived. They are used to provide concrete sample size prescriptions. Certain principal features of these results are that Bayesian asymptotics and asymptotics of the MLE are not the same; Bayes procedures appear to outperform bias uncorrected MLEs; dimensional asymptotics go in the positive direction, i.e., in stead of a curse of dimensionality, one has a gift of dimensionality. Time permitting, simulations and future work will be discussed.