## On the strong consistency, weak limits and practical performance of the ML estimate and Bayesian estimates of a symmetric domain in $$R^k$$.(English)Zbl 1268.62031

Dasgupta, Anirban (ed.), A Festschrift for Herman Rubin. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-61-7/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 45, 291-308 (2004).
Summary: 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: the 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 best. A discussion on the convergence in distribution is also given.
For the entire collection see [Zbl 1066.62002].

### MSC:

 62F10 Point estimation 62F15 Bayesian inference 62F12 Asymptotic properties of parametric estimators 60F05 Central limit and other weak theorems 65C60 Computational problems in statistics (MSC2010)