## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 22, Number 3 (1951), 327-351.

### Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions

#### Abstract

In this paper linear restrictions on regression coefficients are studied. Let the $p \times q_2$ matrix of coefficients of regression of the $p$ dependent variates on $q_2$ of the independent variates be $\mathbf{\bar B}_2$. Maximum likelihood estimates of an $m \times p$ matrix $\Gamma$ satisfying $\Gamma'\mathbf{\bar B}_2 = 0$ and certain other conditions are found under the assumption that the rank of $\mathbf{\bar B}_2$ is $p - m$ and the dependent variates are normally distributed (Section 2). Confidence regions for $\Gamma$ under various conditions are obtained (Section 5). The likelihood ratio test of the hypothesis that the rank of $\mathbf{\bar B}_2$ is a given number is obtained (Section 3). A test of the hypothesis that $\Gamma$ is a certain matrix is given (Section 4). These results are applied to the "$q$-sample problem" (Section 7) and are extended for certain econometric models (Section 6).

#### Article information

**Source**

Ann. Math. Statist., Volume 22, Number 3 (1951), 327-351.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177729580

**Digital Object Identifier**

doi:10.1214/aoms/1177729580

**Mathematical Reviews number (MathSciNet)**

MR42664

**Zentralblatt MATH identifier**

0043.13902

**JSTOR**

links.jstor.org

#### Citation

Anderson, T. W. Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions. Ann. Math. Statist. 22 (1951), no. 3, 327--351. doi:10.1214/aoms/1177729580. https://projecteuclid.org/euclid.aoms/1177729580