## The Annals of Statistics

### Asymptotic Nonnull Distributions for Likelihood Ratio Statistics in the Multivariate Normal Patterned Mean and Covariance Matrix Testing Problem

Ted H. Szatrowski

#### Abstract

The multivariate normal patterned mean and covariance matrix testing problem is studied for general one and $k$-population hypotheses. T. W. Anderson's iterative algorithm for finding the maximum likelihood estimates, the forms of the likelihood ratio tests, and asymptotic chi-square distributions of these tests under the null hypothesis are given. The nonnull asymptotic normal distribution is derived using the standard delta method. This derivation involves using several extensions of matrix identities given in Anderson, matrix derivatives and asymptotic likelihood equations. The forms of the variances are greatly simplified using a result of Szatrowski when the maximum likelihood estimates under the null hypothesis have explicit representations.

#### Article information

Source
Ann. Statist., Volume 7, Number 4 (1979), 823-837.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344732

Digital Object Identifier
doi:10.1214/aos/1176344732

Mathematical Reviews number (MathSciNet)
MR532246

Zentralblatt MATH identifier
0413.62032

JSTOR

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62H15: Hypothesis testing

#### Citation

Szatrowski, Ted H. Asymptotic Nonnull Distributions for Likelihood Ratio Statistics in the Multivariate Normal Patterned Mean and Covariance Matrix Testing Problem. Ann. Statist. 7 (1979), no. 4, 823--837. doi:10.1214/aos/1176344732. https://projecteuclid.org/euclid.aos/1176344732