## The Annals of Statistics

### Semiparametrically efficient rank-based inference for shape. I. optimal rank-based tests for sphericity

#### Abstract

We propose a class of rank-based procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value V0; this includes, for V0={I}k, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so when it must be estimated. The multivariate ranks used throughout are those of the distances—in the metric associated with the null value V0 of the shape matrix—between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31–43]) or, equivalently, with respect to (an extension of ) the test for sphericity introduced by John [Biometrika 59 (1972) 169–173]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performance is investigated via a Monte Carlo study.

#### Article information

Source
Ann. Statist., Volume 34, Number 6 (2006), 2707-2756.

Dates
First available in Project Euclid: 23 May 2007

https://projecteuclid.org/euclid.aos/1179935063

Digital Object Identifier
doi:10.1214/009053606000000731

Mathematical Reviews number (MathSciNet)
MR2329465

Zentralblatt MATH identifier
1114.62066

Subjects
Primary: 62M15: Spectral analysis 62G35: Robustness

#### Citation

Hallin, Marc; Paindaveine, Davy. Semiparametrically efficient rank-based inference for shape. I. optimal rank-based tests for sphericity. Ann. Statist. 34 (2006), no. 6, 2707--2756. doi:10.1214/009053606000000731. https://projecteuclid.org/euclid.aos/1179935063

#### References

• Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley Interscience, Hoboken, NJ.
• Andreou, E. and Werker, B. J. M. (2004). An alternative asymptotic analysis of residual-based statistics. CentER Discussion Paper 2004-56, Tilburg Univ., The Netherlands.
• Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew-normal distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 579–602.
• Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew $t$-distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 367–389.
• Baringhaus, L. (1991). Testing for spherical symmetry of a multivariate distribution. Ann. Statist. 19 899–917.
• Beran, R. (1979). Testing for ellipsoidal symmetry of a multivariate density. Ann. Statist. 7 150–162.
• Bickel, P. J. (1982). On adaptive estimation. Ann. Statist. 10 647–671.
• Bilodeau, M. and Brenner, D. (1999). Theory of Multivariate Statistics. Springer, New York.
• Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862–872.
• Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric tests. Ann. Math. Statist. 29 972–994.
• Falk, M. (2002). The sample covariance is not efficient for elliptical distributions. J. Multivariate Anal. 80 358–377.
• Garel, B. and Hallin, M. (1995). Local asymptotic normality of multivariate ARMA processes with a linear trend. Ann. Inst. Statist. Math. 47 551–579.
• Ghosh, S. K. and Sengupta, D. (2001). Testing for proportionality of multivariate dispersion structures using interdirections. J. Nonparametr. Statist. 13 331–349.
• Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325–346.
• Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, San Diego, CA.
• Hallin, M., Oja, H. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. II. Optimal $R$-estimation of shape. Ann. Statist. 34 2757–2789.
• Hallin, M. and Paindaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103–1133.
• Hallin, M. and Paindaveine, D. (2006). Asymptotic linearity of serial and nonserial multivariate signed rank statistics. J. Statist. Plann. Inference 136 1–32.
• Hallin, M. and Paindaveine, D. (2006). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 327–350.
• Hallin, M. and Werker, B. J. M. (2003). Semiparametric efficiency, distribution-freeness and invariance. Bernoulli 9 137–165.
• Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325.
• Hoeffding, W. (1973). On the centering of a simple linear rank statistic. Ann. Statist. 1 54–66.
• Huynh, H. and Mandeville, G. K. (1979). Validity conditions in repeated-measures designs. Psychological Bull. 86 964–973.
• John, S. (1971). Some optimal multivariate tests. Biometrika 58 123–127.
• John, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika 59 169–173.
• Kac, M. (1939). On a characterization of the normal distribution. Amer. J. Math. 61 726–728.
• Kariya, T. and Eaton, M. L. (1977). Robust tests for spherical symmetry. Ann. Statist. 5 206–215.
• Koltchinskii, V. and Sakhanenko, L. (2000). Testing for ellipsoidal symmetry of a multivariate distribution. In High Dimensional Probability II (E. Giné, D. Mason and J. Wellner, eds.) 493–510. Birkhäuser, Boston.
• Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
• Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics. Some Basic Concepts, 2nd ed. Springer, New York.
• Magnus, J. R. and Neudecker, H. (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics, rev. ed. Wiley, Chichester.
• Marden, J. (1999). Multivariate rank tests. In Multivariate Analysis, Design of Experiments and Survey Sampling (S. Ghosh, ed.) 401–432. Dekker, New York.
• Marden, J. and Gao, Y. (2002). Rank-based procedures for structural hypotheses on covariance matrices. Sankhyā Ser. A 64 653–677.
• Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London.
• Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, Chichester.
• Mauchly, J. W. (1940). Significance test for sphericity of a normal $n$-variate distribution. Ann. Math. Statist. 11 204–209.
• Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. J. Nonparametr. Statist. 5 201–213.
• Muirhead, R. J. and Waternaux, C. M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika 67 31–43.
• Oja, H. (1999). Affine invariant multivariate sign and rank tests and corresponding estimates: A review. Scand. J. Statist. 26 319–343.
• Ollila, E., Croux, C. and Oja, H. (2004). Influence function and asymptotic efficiency of the affine equivariant rank covariance matrix. Statist. Sinica 14 297–316.
• Ollila, E., Hettmansperger, T. P. and Oja, H. (2005). Affine equivariant multivariate sign methods. Preprint, Univ. Jyväskylä.
• Ollila, E., Oja, H. and Croux, C. (2003). The affine equivariant sign covariance matrix: Asymptotic behavior and efficiencies. J. Multivariate Anal. 87 328–355.
• Paindaveine, D. (2006). A Chernoff–Savage result for shape. On the nonadmissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206–2220.
• Puri, M. L. and Sen, P. K. (1985). Nonparametric Methods in General Linear Models. Wiley, New York.
• Randles, R. H. (1982). On the asymptotic normality of statistics with estimated parameters. Ann. Statist. 10 462–474.
• Randles, R. H. (1989). A distribution-free multivariate sign test based on interdirections. J. Amer. Statist. Assoc. 84 1045–1050.
• Schwartz, L. (1973). Théorie des Distributions. Hermann, Paris.
• Sugiura, N. (1972). Locally best invariant test for sphericity and the limiting distributions. Ann. Math. Statist. 43 1312–1316.
• Swensen, A. R. (1985). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. J. Multivariate Anal. 16 54–70.
• Tyler, D. E. (1982). Radial estimates and the test for sphericity. Biometrika 69 429–436.
• Tyler, D. E. (1983). Robustness and efficiency properties of scatter matrices. Biometrika 70 411–420.
• Tyler, D. E. (1987). A distribution-free $M$-estimator of multivariate scatter. Ann. Statist. 15 234–251.
• Tyler, D. E. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika 74 579–589.