On a Characterization of Uniform Distributions

✍ Scribed by S. Dasgupta; A. Goswami; B.V. Rao

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 392 KB
• Volume: 44
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1993.1006

✦ Synopsis

Denoting by (X_{(1,} \leqslant X_{t 2} \leqslant \cdots \leqslant X_{(n)}) the order statistic based on a random sample (X_{1}, X_{2}, \ldots, X_{n}) drawn from a distribution (F), it is shown that the property " (E\left(X_{1} \mid X_{(1)}, X_{(n)}\right)=\frac{1}{2}\left(X_{(1)}+X_{(n)}\right)), almost surely" holds for some (n \geqslant 3) if and only if (F) is either a uniform distribution on an interval or a discrete uniform distribution supported on a set of equispaced points. The heart of our proof is the interesting observation that the above equation involving conditional expectations completely determines the (topological) structure of the support of the undertying distribution (F). ' 1993 Academic Préss. Inc.