## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 3 (1983), 947-958.

### Missing Data in the One-Population Multivariate Normal Patterned Mean and Covariance Matrix Testing and Estimation Problem

#### Abstract

In this paper the multivariate normal linear patterned mean and covariance matrix testing and estimation problem is studied in the presence of missing data for general one-population hypotheses. The Newton-Raphson, Method of Scoring and EM algorithms are given for finding the maximum likelihood estimates. The asymptotic joint distribution of the maximum likelihood estimates under null and alternative hypotheses are derived along with the form of the likelihood ratio statistic and its asymptotically Chi squared null and asymptotically normal nonnull distributions. The distributions of the maximum likelihood estimates and nonnull distributions of the likelihood ratio tests are derived using the standard multivariate and univariate delta method respectively, and may be evaluated at a parameter point under the alternative hypothesis parameter space or at a parameter point in a parameter space that contains the null and alternative hypothesis parameter spaces. New results for these problems in the presence of complete data as well as known results (Szatrowski, 1979) are special cases of the results of this paper.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 3 (1983), 947-958.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346260

**Digital Object Identifier**

doi:10.1214/aos/1176346260

**Mathematical Reviews number (MathSciNet)**

MR707944

**Zentralblatt MATH identifier**

0524.62050

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H10: Distribution of statistics

Secondary: 62H15: Hypothesis testing

**Keywords**

Asymptotic efficiency asymptotic nonnull distribution delta method EM algorithm hypothesis testing iterative computations matrix derivatives maximum likelihood estimates missing data patterned means patterened covariance matrices

#### Citation

Szatrowski, Ted H. Missing Data in the One-Population Multivariate Normal Patterned Mean and Covariance Matrix Testing and Estimation Problem. Ann. Statist. 11 (1983), no. 3, 947--958. doi:10.1214/aos/1176346260. https://projecteuclid.org/euclid.aos/1176346260