Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series A, Pt. 1 ,pp.145-149

ON A CHARACTERIZATION OF THE NORMAL DISTRIBUTION BY A PROPERTY OF ORDER STATISTICS

By

JIAN-LUN XU,
* University of Houston, Houston*

*SUMMARY.* Let *X*_{(n)}-*X*_{(1)} be the range of an iid
sample **X**=(*X*_{1}, ..., *X*_{n})^{'} with a continuous density
function *f* and let
*Y*_{(n)}-*Y*_{(1)} be the range of
**Y**=G**X**, where
G is any *n* *x* n orthogonal matrix.
Klebanov (1973) proved that
$X_1$ is normally distributed if and only if *X*_{(n)}-*X*_{(1)} and
*Y*_{(n)}-*Y*_{(1)}
have the same distribution provided *n*>=4.
The objective of
this note is to show that
Klebanov's (1973) result is true when *n*>=2.

*AMS (1991) subject classification.* Primary 62E10; secondary 62E15.

*Key words and phrases. *Normal characterization; order statistics; range.