The Annals of Mathematical Statistics

Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions

T. W. Anderson

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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


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