## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 1 (1974), 109-117.

### Asymptotic Expansions of the Non-Null Distributions of Likelihood Ratio Criteria for Covariance Matrices

C. G. Khatri and M. S. Srivastava

#### Abstract

In this paper, asymptotic expansions of the non-null distributions of the likelihood ratio criteria are obtained for testing the hypotheses: (a) $H_1: \Sigma = \sigma^2I, \sigma^2 > 0$, (b) $H_2: \Sigma_1 = \Sigma_2$, against alternatives which are close to the hypothesis. These expansions are of chi-square type. The first problem has been considered by Sugiura (1969) but because of the singularity at the hypothesis, his expansion will not be good for alternatives close to the hypothesis. The second problem has been considered by de Waal (1970) but the expansion given by him is invalid.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 1 (1974), 109-117.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342617

**Digital Object Identifier**

doi:10.1214/aos/1176342617

**Mathematical Reviews number (MathSciNet)**

MR345317

**Zentralblatt MATH identifier**

0296.62037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H10: Distribution of statistics

Secondary: 62E20: Asymptotic distribution theory 62H15: Hypothesis testing

**Keywords**

Asymptotic non-null distributions likelihood ratio tests sphericity equality of covariances chi-square distribution

#### Citation

Khatri, C. G.; Srivastava, M. S. Asymptotic Expansions of the Non-Null Distributions of Likelihood Ratio Criteria for Covariance Matrices. Ann. Statist. 2 (1974), no. 1, 109--117. doi:10.1214/aos/1176342617. https://projecteuclid.org/euclid.aos/1176342617