Skip to main content
SpringerLink
Log in

Robust asymptotic tests for the equality of multivariate coefficients of variation

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

In order to easily compare several populations on the basis of more than one feature, multivariate coefficients of variation (MCV) may be used as they allow to summarize relative dispersion in a single index. However, up to date, no test of equality of one or more MCVs has been developed in the literature. In this paper, several classical and robust Wald-type tests are proposed and studied. The asymptotic distributions of the test statistics are derived under elliptical symmetry, and the asymptotic efficiency of the robust versions is compared to the classical tests. Robustness of the proposed procedures is examined through partial and joint influence functions of the test statistic, as well as by means of power and level influence functions. A simulation study compares the performance of the classical and robust tests under uncontaminated and contaminated schemes, and the difference with the usual covariance homogeneity test is highlighted. As a by-product, these tests may also be considered in the univariate context where they yield procedures that are both robust and easy-to-use. They provide an interesting alternative to the numerous parametric tests existing in the literature, which are, in most cases, unreliable in presence of outliers. The methods are illustrated on a real data set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. At distribution F, the weight function was set to \(I_{[0,q_{\delta }]}\) with \(\delta =0.025\) and \(q_{\delta }= G^{-1}(1-\delta )\) where \(G(t)={\mathbb {P}}_{F}[\mathbf {X}^t\mathbf {X}\le t]\).

References

  • Aerts S, Haesbroeck G, Ruwet C (2015) Multivariate coefficients of variation: comparison and influence functions. J Multivar Anal 142:183–198

    Article  MathSciNet  MATH  Google Scholar 

  • Aerts S, Haesbroeck G, Ruwet C (2016) Distribution under elliptical symmetry of a distance-based multivariate coefficient of variation. Stat Pap. doi:10.1007/s00362-016-0777-4

  • Albert A, Zhang L (2010) A novel definition of the multivariate coefficient of variation. Biom J 52:667–675

    Article  MathSciNet  MATH  Google Scholar 

  • Bartlett MS (1937) Properties of sufficiency and statistical tests. J R Soc A Math 160:268–282

    MATH  Google Scholar 

  • Basu A, Mandal A, Martin A, Pardo L (2015) Generalized Wald-type tests based on minimum density power divergence estimators. Statistics 50:1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Bennett BM (1976) On an approximate test for homogeneity of coefficients of variation. In: Ziegler WJ (ed) Contribution to applied statistics. Birkhauser Verlag, Basel and Stuttgart, pp 169–171

    Chapter  Google Scholar 

  • Cator EA, Lopuhäa HP (2010) Asymptotic expansion of the minimum covariance determinant estimators. J Multivar Anal 101:2372–2388

    Article  MathSciNet  MATH  Google Scholar 

  • Croux C, Haesbroeck G (1999) Influence function and efficiency of the minimum covariance determinant estimator. J Multivar Anal 71:161–190

    Article  MathSciNet  MATH  Google Scholar 

  • Falk M (1997) Asymptotic independance of median and MAD. Stat Probab Lett 34:341–345

    Article  MATH  Google Scholar 

  • Feltz CJ, Miller GE (1996) An asymptotic test for the equality of coefficients of variation. Stat Med 15:647–658

    Article  Google Scholar 

  • Forkman J (2009) Estimator and tests for common coefficients of variation in normal distributions. Commun Stat Theory 38:233–251

    Article  MathSciNet  MATH  Google Scholar 

  • Garcia-Perez A (2012) A linear approximation to the power function of a test. Metrika 75:855–875

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh A, Mandal A, Martin N, Pardo L (2016) Influence analysis of robust Wald-type tests. J Multivar Anal 147:102–126

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh A, Basu A (2013) Robust estimation for independent non-homogeneous observations using density power divergence with applications to linear regression. Electron J Stat 7:2420–2456

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta CR, Ma S (1996) Testing the equality of coefficients of variation in \(K\) normal populations. Commun Stat Theory 25:115–132

    Article  MathSciNet  MATH  Google Scholar 

  • Hallin M, Paindaveine D (2009) Optimal tests for homogeneity of covariance, scale, and shape. J Multivar Anal 100:422–444

    Article  MathSciNet  MATH  Google Scholar 

  • Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, New York

    MATH  Google Scholar 

  • Healy MJ (1979) Outliers in clinical chemistry quality control schemes. Clin Chem 25:675–677

    Google Scholar 

  • Heritier S, Ronchetti EM (1994) Robust bounded-influence tests in general parametric models. JASA 89:897–904

    Article  MathSciNet  MATH  Google Scholar 

  • Krishnamoorthy K, Yu J (2004) Modified Nel and van der Merwe Test for the Multivariate Behrens-Fisher Problem. Stat Probab Lett 66:161–169

    Article  MathSciNet  MATH  Google Scholar 

  • Krishnamoorthy K, Lee M (2014) Improved tests for the equality of normal coefficients of variation. Comput Stat 29:215–232

    Article  MathSciNet  MATH  Google Scholar 

  • Ledoit O, Wolf M (2008) Robust performance hypothesis testing with the Sharpe ratio. J Empir Financ 15:850–859

    Article  Google Scholar 

  • Loh W-Y (1984) Bounds on AREs for restricted classes of distributions defined via tail-orderings. Ann Stat 12:685–701

    Article  MathSciNet  MATH  Google Scholar 

  • Lopuhäa HP (1997) Asymptotic expansion of S estimators of location and covariance. Stat Neerl 51:220–237

    Article  MathSciNet  MATH  Google Scholar 

  • Lopuhäa HP (1999) Asymptotics of reweighted estimators of multivariate location and scatter. Ann Stat 27:1638–1665

    Article  MathSciNet  MATH  Google Scholar 

  • McKay A (1932) Distribution of the coefficient of variation and the extended \(t\)-distribution. J R Stat Soc 95:695–698

    Article  MATH  Google Scholar 

  • Nairy KS, Rao KA (2003) Tests of coefficients of variation of normal population. Commun Stat Simul 32:641–661

    Article  MathSciNet  MATH  Google Scholar 

  • Pardo MC, Pardo JA (2000) Use of Rényi’s divergence to test for the equality of the coefficients of variation. J Comput Appl Math 116:93–104

    Article  MathSciNet  MATH  Google Scholar 

  • Pires AM, Branco JA (2002) Partial influence functions. J Multivar Anal 83:451–468

    Article  MathSciNet  MATH  Google Scholar 

  • Puri ML, Sen PK (1971) Nonparametric methods in multivariate analysis. Wiley, New York

    MATH  Google Scholar 

  • Rousseeuw PJ, Ronchetti EM (1981) Influence curve for general statistics. J Comput Appl Math 7:161–166

    Article  MathSciNet  MATH  Google Scholar 

  • Schott JR (2001) Some tests for the equality of covariance matrices. J Stat Plan Inference 94:25–36

    Article  MathSciNet  MATH  Google Scholar 

  • Tsou TS (2009) A robust score test for testing several coefficients of variation with unknown underlying distributions. Commun Stat Theory 38:1350–1360

    Article  MathSciNet  MATH  Google Scholar 

  • Verrill SP, Johnson RA (2007) Confidence bounds and hypothesis tests for normal distribution coefficients of variation. Research Paper FPL-RP-638. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory

  • Voinov VG, Nikulin MS (1996) Unbiased estimators and their applications multivariate case, vol 2. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Zhang L, Albarède S, Dumont G, Van Campenhout C, Libeer J, Albert A (2010) The multivariate coefficient of variation for comparing serum protein electrophoresis techniques in external quality assessment schemes. Accredit Qual Assur 15:351–357

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to express their thanks to Professor A. Albert (School of Public Health, University of Liege) for making the EQA data available. This work was partially supported by the IAP Research Network P7/06 of the Belgian State. They also thank the referees for providing constructive comments and helpful suggestions.

Author information

Authors and Affiliations

  1. HEC-ULg, University of Liege (ULg, N1), Rue Louvrex 14, 4000, Liège, Belgium

    Stephanie Aerts

  2. University of Liege (ULg, Polytech 1), Allée de la Découverte 12, 4000, Liège, Belgium

    Gentiane Haesbroeck

Authors
  1. Stephanie Aerts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gentiane Haesbroeck
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephanie Aerts.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 36 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aerts, S., Haesbroeck, G. Robust asymptotic tests for the equality of multivariate coefficients of variation. TEST 26, 163–187 (2017). https://doi.org/10.1007/s11749-016-0504-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-016-0504-4

Keywords

Mathematics Subject Classification

Search

Navigation