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September, 1951 Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions
T. W. Anderson
Ann. Math. Statist. 22(3): 327-351 (September, 1951). DOI: 10.1214/aoms/1177729580

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).

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T. W. Anderson. "Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions." Ann. Math. Statist. 22 (3) 327 - 351, September, 1951. https://doi.org/10.1214/aoms/1177729580

Information

Published: September, 1951
First available in Project Euclid: 28 April 2007

zbMATH: 0043.13902
MathSciNet: MR42664
Digital Object Identifier: 10.1214/aoms/1177729580

Rights: Copyright © 1951 Institute of Mathematical Statistics

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Vol.22 • No. 3 • September, 1951
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