## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 4 (1983), 1243-1250.

### A Class of Asymptotic Tests for Principal Component Vectors

#### Abstract

In this paper, the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered. A class of invariant asymptotic tests based on the sample covariance matrix is derived. Tests in this class are shown to be consistent and their local power functions are given. The arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix. The results are shown to generalize when the procedures are based on any affine-invariant $M$-estimate of scatter and when the population is elliptical.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 4 (1983), 1243-1250.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346337

**Digital Object Identifier**

doi:10.1214/aos/1176346337

**Mathematical Reviews number (MathSciNet)**

MR720269

**Zentralblatt MATH identifier**

0544.62053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H15: Hypothesis testing

Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62H25: Factor analysis and principal components; correspondence analysis 62E20: Asymptotic distribution theory

**Keywords**

Elliptical distributions invariance noncentral Wishart robustness spectral decomposition

#### Citation

Tyler, David E. A Class of Asymptotic Tests for Principal Component Vectors. Ann. Statist. 11 (1983), no. 4, 1243--1250. doi:10.1214/aos/1176346337. https://projecteuclid.org/euclid.aos/1176346337