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September, 1952 The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics
Eugene Lukacs
Ann. Math. Statist. 23(3): 442-449 (September, 1952). DOI: 10.1214/aoms/1177729389

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.

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Eugene Lukacs. "The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics." Ann. Math. Statist. 23 (3) 442 - 449, September, 1952. https://doi.org/10.1214/aoms/1177729389

Information

Published: September, 1952
First available in Project Euclid: 28 April 2007

zbMATH: 0047.38102
MathSciNet: MR50228
Digital Object Identifier: 10.1214/aoms/1177729389

Rights: Copyright © 1952 Institute of Mathematical Statistics

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Vol.23 • No. 3 • September, 1952
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