The Annals of Mathematical Statistics

The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics

Eugene Lukacs

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Abstract

The following theorem is proved. If a univariate distribution has moments of first and second order and admits a homogeneous and symmetric quadratic statistic $Q$ which is independently distributed of the mean of a sample of $n$ drawn from this distribution, then it is either the normal distribution ($Q$ is then proportional to the variance) or the degenerate distribution (in this case no restriction is imposed on $Q$) or a step function with two symmetrically located steps (in this case $Q$ is the sum of the squared observations). The converse of this statement is also true.

Article information

Source
Ann. Math. Statist. Volume 23, Number 3 (1952), 442-449.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aoms/1177729389

Digital Object Identifier
doi:10.1214/aoms/1177729389

Mathematical Reviews number (MathSciNet)
MR50228

Zentralblatt MATH identifier
0047.38102

JSTOR
links.jstor.org

Citation

Lukacs, Eugene. The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics. Ann. Math. Statist. 23 (1952), no. 3, 442--449. doi:10.1214/aoms/1177729389. http://projecteuclid.org/euclid.aoms/1177729389.


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