×

A generalized maximum likelihood characterization of the normal distribution. (English) Zbl 0664.62010

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 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 F. W. Scholz [Can. J. Stat. 8, 193-203 (1980; Zbl 0466.62006)].

MSC:

62E10 Characterization and structure theory of statistical distributions
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Findeisen P (1982) Die Charakterisierung der Normalverteilung nach Gauß. Metrika 29:55–64 · Zbl 0493.62013 · doi:10.1007/BF01893364
[2] Natanson IP (1980) Theory of functions of a real variable, 2 vols. Unger, New York
[3] Scholz FW (1980) Towards a unified definition of maximum likelihood. Canad J Statist 8:193–203 · Zbl 0466.62006 · doi:10.2307/3315231
[4] Teicher H (1961) Maximum likelihood characterization of distributions. Ann Math Statist 32:1214–1222 · Zbl 0102.14702 · doi:10.1214/aoms/1177704861
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.