## Annals of Probability

- Ann. Probab.
- Volume 14, Number 2 (1986), 696-701.

### Sphericity and the Normal Law

#### Abstract

Let $\mathbf{x} = (x_1,\cdots, x_n)'$ be a random vector in $R^n$. Two characterizations of normality are given. One involves the existence of two linear combinations of the $\{x_j\}$ that are independent in every coordinate system. The other, which is actually a consequence of the first, assumes that $\mathbf{x}$ obeys a linear model with spherical errors and involves sufficiency of the least-squares estimator.

#### Article information

**Source**

Ann. Probab., Volume 14, Number 2 (1986), 696-701.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992538

**Digital Object Identifier**

doi:10.1214/aop/1176992538

**Mathematical Reviews number (MathSciNet)**

MR832031

**Zentralblatt MATH identifier**

0595.60016

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E99: None of the above, but in this section

Secondary: 62B99: None of the above, but in this section

**Keywords**

Characterization of normality sphericity sufficiency least-squares estimator

#### Citation

Berk, Robert H. Sphericity and the Normal Law. Ann. Probab. 14 (1986), no. 2, 696--701. doi:10.1214/aop/1176992538. https://projecteuclid.org/euclid.aop/1176992538