## 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

#### 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**

https://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. https://projecteuclid.org/euclid.aoms/1177729389