Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions (original) (raw)
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 ptimesq_2p \times q_2ptimesq2 matrix of coefficients of regression of the ppp dependent variates on q2q_2q2 of the independent variates be mathbfbarB2\mathbf{\bar B}_2mathbfbarB2. Maximum likelihood estimates of an mtimespm \times pmtimesp matrix Gamma\GammaGamma satisfying Gamma′mathbfbarB2=0\Gamma'\mathbf{\bar B}_2 = 0Gamma′mathbfbarB2=0 and certain other conditions are found under the assumption that the rank of mathbfbarB2\mathbf{\bar B}_2mathbfbarB2 is p−mp - mp−m and the dependent variates are normally distributed (Section 2). Confidence regions for Gamma\GammaGamma under various conditions are obtained (Section 5). The likelihood ratio test of the hypothesis that the rank of mathbfbarB2\mathbf{\bar B}_2mathbfbarB_2 is a given number is obtained (Section 3). A test of the hypothesis that Gamma\GammaGamma 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).
Citation
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
Digital Object Identifier: 10.1214/aoms/1177729580
Rights: Copyright © 1951 Institute of Mathematical Statistics
Vol.22 • No. 3 • September, 1951
