Open Access
December 2020 A necessary test for elliptical symmetry based on the uniform distribution over the Stiefel manifold
Toshiya Iwashita, Bernhard Klar
Author Affiliations +
SUT J. Math. 56(2): 129-145 (December 2020). DOI: 10.55937/sut/1609967403


This paper provides a new procedure for testing the null hypothesis of multivariate elliptical symmetry. A test for uniformity over the Stiefel manifold based on modified degenerate V-statistics is employed since the test statistic proposed in this paper consists of independent random matrices, formed by the scaled residuals (or the Studentized residuals), which are uniformly distributed over the Stiefel manifold under the null hypothesis. Also, Monte Carlo simulation studies are carried out to evaluate the type I error and power of the test. Finally, the procedure is illustrated using the Iris data.

Funding Statement

The first author has been partially supported by Grant-in-Aid for Scientific Research(C), JSPS KAKENHI Grand Numbers 18K11198, 18K03428.


The authors wish to express their thanks to an anonymous reviewer for valuable comments and suggestions which greatly improved a previous version of this paper, in particular the original proofs of Lemma 2.1 and Theorem 2.2.


Download Citation

Toshiya Iwashita. Bernhard Klar. "A necessary test for elliptical symmetry based on the uniform distribution over the Stiefel manifold." SUT J. Math. 56 (2) 129 - 145, December 2020.


Received: 14 April 2020; Published: December 2020
First available in Project Euclid: 8 June 2022

Digital Object Identifier: 10.55937/sut/1609967403

Primary: 62H10 , 62H15

Keywords: elliptical distribution , left-spherical distribution , scaled residuals , spherical distribution , Stiefel manifold , uniform distribution

Rights: Copyright © 2020 Tokyo University of Science

Vol.56 • No. 2 • December 2020
Back to Top