## The Annals of Statistics

### Asymptotic distribution of the reduced rank regression estimator under general conditions

T. W. Anderson

#### Abstract

In the regression model $\mathbf{Y} = \eta + \mathbf{BX} + \mathbf{Z}$ with $\mathbf{Z}$ unobserved, $\mathscr{E}\mathbf{Z} = \mathbf{0}$ and $\mathscr{E}\mathbf{ZZ}' = \mathbf{\Sigma}_{ZZ}$, the least squares estimator of $\mathbf{B}$ is $\hat{\mathbf{B}} = \mathbf{S}_{YX}\mathbf{S}_{XX}^{-1}$. If the rank of $\mathbf{B}$ is known to be $k$ less than the dimensions of $\mathbf{Y}$ and $\mathbf{X}$, the reduced rank regression estimator of $\mathbf{B}$, say $\mathbf{B}_k$, depends on the first $k$ canonical variates of $\mathbf{Y}$ and $\mathbf{X}$. The asymptotic distribution of $\hat{\mathbf{B}}_k$ is obtained and compared with the asymptotic distribution of $\hat{\mathbf{B}}$. The advantage of $\hat{\mathbf{B}}_k$ is characterized.

#### Article information

Source
Ann. Statist., Volume 27, Number 4 (1999), 1141-1154.

Dates
First available in Project Euclid: 4 April 2002

https://projecteuclid.org/euclid.aos/1017938918

Digital Object Identifier
doi:10.1214/aos/1017938918

Mathematical Reviews number (MathSciNet)
MR1740118

Zentralblatt MATH identifier
0961.62011

Subjects
Primary: 62H10: Distribution of statistics 62E20: Asymptotic distribution theory
Secondary: 62H12: Estimation

#### Citation

Anderson, T. W. Asymptotic distribution of the reduced rank regression estimator under general conditions. Ann. Statist. 27 (1999), no. 4, 1141--1154. doi:10.1214/aos/1017938918. https://projecteuclid.org/euclid.aos/1017938918

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• STANFORD, CALIFORNIA 94305-4065 E-MAIL: twa@stat.stanford.edu