A generalized maximum likelihood characterization of the normal distribution (Q1115050)

Let P be a probability measure on \({\mathbb{R}}\) and \({\mathcal I}_ x\) be the set of all n-dimensional rectangles containing x. If for all \(x\in {\mathbb{R}}^ n\) and \(\theta\in {\mathbb{R}}\) the inequality \[ \liminf_{{\mathcal I}_ x\ni I\downarrow \{x\}}P^ n(I-\bar x)/P^ n(I-\theta)\geq 1 \] holds, P is a normal distribution with mean 0 or the unit mass at 0. The result generalizes \textit{H. Teicher}'s [Ann. Math. Stat. 32, 1214-1222 (1961; Zbl 0102.147)] maximum likelihood characterization of the normal density to a characterization of \(N(0,\sigma^ 2)\) among all distributions (including those without density). The m.l. principle used is that of \textit{F. W. Scholz} [Can. J. Stat. 8, 193-203 (1980; Zbl 0466.62006)].