Optimal tests for homogeneity of covariance, scale, and shape
AbstractThe assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic context of multivariate Gaussian models. And the modified likelihood ratio tests (MLRT) that are used in everyday practice are known to be highly sensitive to violations of Gaussian assumptions. In this paper, we provide a complete and systematic study of the problem, and propose test statistics which, while preserving the optimality features of the MLRT under multinormal assumptions, remain valid under unspecified elliptical densities with finite fourth-order moments. As a first step, the Le Cam LAN approach is used for deriving locally and asymptotically optimal testing proceduresÂ for any specified m-tuple of radial densities f=(f1,...,fm). Combined with an estimation of the m densities f1,...,fm, these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudo-Gaussian tests-namely, tests that are locally and asymptotically optimal at Gaussian densities while remaining valid under a much broader class of distributions-in general are preferable. We therefore construct two pseudo-Gaussian modifications of the Gaussian version of the optimal test . The first one, , is valid under the class of homokurtic m-tuples f, while the validity of the second,Â , extends to the heterokurtic ones, that is, to arbitrary m-tuples of elliptical distributions with finite fourth-order moments. We moreover show that these tests are asymptotically equivalent to modified Wald tests recently proposed by Schott [J.R. Schott, Some tests for the equality of covariance matrices, Journal of Statistical Planning and Inference 94 (2001) 25-36]. This settles the optimality properties of the latter. Our results however are much more informative than Schott's. They also allow for computing local powers, and for an ANOVA-type decomposition of the test statistics into two mutually independent parts providing tests against subalternatives of scale and shape heterogeneity, respectively, thus supplying additional insight into the reasons why rejection occurs. Reinforcing a result of Yanagihara etÂ al.Â [H. Yanagihara, T. Tonda, C. Matsumoto, The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption, Journal of Multivariate Analysis 96 (2005) 237-264], we further show why another approach, based on bootstrapped critical values of the Gaussian MLRT statistic, although producing asymptotically valid pseudo-Gaussian tests, is highly unsatisfactory in this context. We also develop optimal pseudo-Gaussian tests for scale homogeneity and for shape homogeneity, based on the same methodology. Finally, the small-sample properties of the proposed procedures are investigated via a Monte-Carlo study.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Bibliographic InfoArticle provided by Elsevier in its journal Journal of Multivariate Analysis.
Volume (Year): 100 (2009)
Issue (Month): 3 (March)
Contact details of provider:
Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description
Find related papers by JEL classification:
- 62M - - - - - -
- 62G - - - - - -
- Ell - Macroeconomics and Monetary Economics - - - - -
- den - - - - - -
- Hom - Public Economics - - - - -
- of - - -
- cov - - - - - -
- Loc - Industrial Organization - - - - -
- asy - - - - - -
- nor - - - - - -
- Loc - Industrial Organization - - - - -
- asy - - - - - -
- mos - - - - - -
- str - - - - - -
- tes - - - - - -
- Mul - Business Administration and Business Economics; Marketing; Accounting - - - - -
- ana - - - - - -
- of - - -
- var - - - - - -
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Taskinen, Sara & Croux, Christophe & Kankainen, Annaliisa & Ollila, Esa & Oja, Hannu, 2006. "Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 359-384, February.
- Liebscher, Eckhard, 2005. "A semiparametric density estimator based on elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 205-225, January.
- Arjun Gupta & Jin Xu, 2006. "On Some Tests of the Covariance Matrix Under General Conditions," Annals of the Institute of Statistical Mathematics, Springer, vol. 58(1), pages 101-114, March.
- Lutz Dümbgen & David E. Tyler, 2005. "On the Breakdown Properties of Some Multivariate M-Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 32(2), pages 247-264.
- Carter, E. M. & Srivastava, M. S., 1977. "Monotonicity of the power functions of modified likelihood ratio criterion for the homogeneity of variances and of the sphericity test," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 229-233, March.
- Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
- Yanagihara, Hirokazu & Tonda, Tetsuji & Matsumoto, Chieko, 2005. "The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 237-264, October.
- Srivastava, M. S. & Khatri, C. G. & Carter, E. M., 1978. "On monotonicity of the modified likelihood ratio test for the equality of two covariances," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 262-267, June.
- Hallin Marc & Paindaveine Davy, 2006. "Parametric and semiparametric inference for shape: the role of the scale functional," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 24, December.
- Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
- Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2011. "Optimal Rank-Based Tests for Common Principal Components," Working Papers ECARES ECARES 2011-032, ULB -- Universite Libre de Bruxelles.
- Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
- Marc Hallin, 2008. "On the Non Gaussian Asymptotics of the Likelihood Ratio Test Statistic for Homogeneity of Covariance," Working Papers ECARES 2008_039, ULB -- Universite Libre de Bruxelles.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Zhang, Lei).
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.