Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 2, 193--205

CANONICAL CORRELATIONS AND VARIATES, REDUCED RANK REGRESSION, MAXIMUM LIKELIHOOD ESTIMATORS, TEST OF RANK

By

T.W. Anderson, Stanford University, USA

SUMMARY. In the classical linear regression model with $p$ dependent variables constituting the vector $\bY$ and $q$ independent variables constituting the vector $\bX$ the rank $k$ of the regression matrix $\bB$ of $\bY$ on $\bX$ may be less than $p$ and $q$. In that case the estimator of $\bB$, called the ``reduced rank regression estimator,'' is composed of the $k$ canonical variables corresponding to the $k$ largest canonical correlations (Anderson, 1951). The asymptotic distribution of this estimator has been found when the rank is correctly specified and $\bX$ and the residual $\bY - \bB\bX$ are independent with finite variances (Anderson, 1999b). The reduced rank regression estimator is more efficient than the least squares estimator, markedly so if $k$ is small. This paper considers the properties of the estimator when the rank is not correctly specified. When the specified rank of $\bB$ is less than the true rank, biases occur; when the specified rank is greater, variances of estimators and predictors are unnecessarily large. These results are related to tests concerning the rank.

*AMS (1991) subject classification}. *Primary 62H12; secondary 62J05.

*Key words and phrases. *Canonical correlations and variates, reduced rank regression, maximum likelihood estimators, test of rank.