## The Annals of Statistics

### Asymptotic Theory for Common Principal Component Analysis

Bernard N. Flury

#### Abstract

Under the common principal component model $k$ covariance matrices $\mathbf{\Sigma}_1,\cdots,\mathbf{\Sigma}_k$ are simultaneously diagonalizable, i.e., there exists an orthogonal matrix $\mathbf{\beta}$ such that $\mathbf{\beta'\Sigma_i\beta = \Lambda_i}$ is diagonal for $i = 1,\cdots, k$. In this article we give the asymptotic distribution of the maximum likelihood estimates of $\mathbf{\beta}$ and $\mathbf{\Lambda}_i$. Using these results, we derive tests for (a) equality of eigenvectors with a given set of orthonormal vectors, and (b) redundancy of $p - q$ (out of $p$) principal components. The likelihood-ratio test for simultaneous sphericity of $p - q$ principal components in $k$ populations is derived, and some of the results are illustrated by a biometrical example.

#### Article information

Source
Ann. Statist., Volume 14, Number 2 (1986), 418-430.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349930

Digital Object Identifier
doi:10.1214/aos/1176349930

Mathematical Reviews number (MathSciNet)
MR840506

Zentralblatt MATH identifier
0613.62075

JSTOR