The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 23, Number 3 (1952), 442-449.
The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics
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.
Ann. Math. Statist. Volume 23, Number 3 (1952), 442-449.
First available in Project Euclid: 28 April 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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. https://projecteuclid.org/euclid.aoms/1177729389