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=(X1, ..., Xn)' with a continuous density function f and let Y(n)-Y(1) be the range of Y=GX, 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.

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