Centre for Mathematical Physics and Stochastics

Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-RR 2000-34

September 2000

Let $Z_{1},Z_{2},...$ be a sequence of i.i.d. $\mathbb{R}^{k}$-valued random vectors distributed uniformly on a star-shaped set $A_{0}$ belonging to a certain known class $\mathcal{A}$ of star-shaped sets satisfying some regularity assumptions. For $A_{0}$ we consider the maximum likelihood estimator $\hat{A}_{n}$ which minimizes the volume among the sets from $mathcal{A}$ covering the whole sample $Z_{1},Z_{2},...$. The purpose of the paper is to investigate the asymptotic properties of $\hat{A}_{n}$. We prove a functional limit theorem for the radius-vector functions of $\hat{A}_{n}$. In particular we show that $n \lambda(\hat{A}_{n} \triangle A_{0})$ converges in distribution and we find the limit.

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