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Semiparametrically efficient rank-based inference for shape. I: optimal rank-based tests for sphericity. (English) Zbl 1114.62066

Summary: 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 \(V_0\); this includes, for \(V_0=I_k\), 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 \(V_o\) 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 R. J. Muirhead and C. M. Waternaux [Biometrika 67, 31–43 (1980; Zbl 0448.62037)] or, equivalently, with respect to (an extension of) the test for sphericity introduced by S. John [ibid. 59, 169–173 (1972; Zbl 0231.62072)]. 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.

MSC:

62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62H10 Multivariate distribution of statistics
65C05 Monte Carlo methods
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[1] Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis , 3rd ed. Wiley Interscience, Hoboken, NJ. · Zbl 1039.62044
[2] 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.
[3] 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. JSTOR: · Zbl 0924.62050
[4] 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. JSTOR: · Zbl 1065.62094
[5] Baringhaus, L. (1991). Testing for spherical symmetry of a multivariate distribution. Ann. Statist. 19 899–917. · Zbl 0725.62053
[6] Beran, R. (1979). Testing for ellipsoidal symmetry of a multivariate density. Ann. Statist. 7 150–162. · Zbl 0406.62029
[7] Bickel, P. J. (1982). On adaptive estimation. Ann. Statist . 10 647–671. · Zbl 0489.62033
[8] Bilodeau, M. and Brenner, D. (1999). Theory of Multivariate Statistics . Springer, New York. · Zbl 0930.62054
[9] Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862–872. JSTOR: · Zbl 0869.62040
[10] Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric tests. Ann. Math. Statist. 29 972–994. · Zbl 0092.36501
[11] Falk, M. (2002). The sample covariance is not efficient for elliptical distributions. J. Multivariate Anal. 80 358–377. · Zbl 0998.62052
[12] Garel, B. and Hallin, M. (1995). Local asymptotic normality of multivariate ARMA processes with a linear trend. Ann. Inst. Statist. Math. 47 551–579. · Zbl 0841.62076
[13] Ghosh, S. K. and Sengupta, D. (2001). Testing for proportionality of multivariate dispersion structures using interdirections. J. Nonparametr. Statist. 13 331–349. · Zbl 1008.62061
[14] Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325–346. · Zbl 0187.16401
[15] Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests , 2nd ed. Academic Press, San Diego, CA.
[16] 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. · Zbl 1115.62059
[17] Hallin, M. and Paindaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103–1133. · Zbl 1101.62348
[18] Hallin, M. and Paindaveine, D. (2006). Asymptotic linearity of serial and nonserial multivariate signed rank statistics. J. Statist. Plann. Inference 136 1–32. · Zbl 1082.62049
[19] Hallin, M. and Paindaveine, D. (2006). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 327–350. · Zbl 1111.62002
[20] Hallin, M. and Werker, B. J. M. (2003). Semiparametric efficiency, distribution-freeness and invariance. Bernoulli 9 137–165. · Zbl 1020.62042
[21] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 293–325. · Zbl 0032.04101
[22] Hoeffding, W. (1973). On the centering of a simple linear rank statistic. Ann. Statist. 1 54–66. · Zbl 0255.62015
[23] Huynh, H. and Mandeville, G. K. (1979). Validity conditions in repeated-measures designs. Psychological Bull. 86 964–973.
[24] John, S. (1971). Some optimal multivariate tests. Biometrika 58 123–127. JSTOR: · Zbl 0218.62055
[25] John, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika 59 169–173. JSTOR: · Zbl 0231.62072
[26] Kac, M. (1939). On a characterization of the normal distribution. Amer. J. Math. 61 726–728. JSTOR: · Zbl 0022.06102
[27] Kariya, T. and Eaton, M. L. (1977). Robust tests for spherical symmetry. Ann. Statist. 5 206–215. · Zbl 0361.62033
[28] 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. · Zbl 0958.62056
[29] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory . Springer, New York. · Zbl 0605.62002
[30] Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics. Some Basic Concepts , 2nd ed. Springer, New York. · Zbl 0952.62002
[31] Magnus, J. R. and Neudecker, H. (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics , rev. ed. Wiley, Chichester. · Zbl 0912.15003
[32] Marden, J. (1999). Multivariate rank tests. In Multivariate Analysis , Design of Experiments and Survey Sampling (S. Ghosh, ed.) 401–432. Dekker, New York. · Zbl 0946.62060
[33] Marden, J. and Gao, Y. (2002). Rank-based procedures for structural hypotheses on covariance matrices. Sankhyā Ser. A 64 653–677. · Zbl 1192.62152
[34] Mardia, K. V. (1972). Statistics of Directional Data . Academic Press, London. · Zbl 0244.62005
[35] Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics . Wiley, Chichester. · Zbl 0935.62065
[36] Mauchly, J. W. (1940). Significance test for sphericity of a normal \(n\)-variate distribution. Ann. Math. Statist. 11 204–209. · Zbl 0023.24703
[37] Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. J. Nonparametr. Statist. 5 201–213. · Zbl 0857.62056
[38] 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. JSTOR: · Zbl 0448.62037
[39] Oja, H. (1999). Affine invariant multivariate sign and rank tests and corresponding estimates: A review. Scand. J. Statist. 26 319–343. · Zbl 0938.62063
[40] 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. · Zbl 1035.62044
[41] Ollila, E., Hettmansperger, T. P. and Oja, H. (2005). Affine equivariant multivariate sign methods. Preprint, Univ. Jyväskylä. · Zbl 1090.62052
[42] Ollila, E., Oja, H. and Croux, C. (2003). The affine equivariant sign covariance matrix: Asymptotic behavior and efficiencies. J. Multivariate Anal. 87 328–355. · Zbl 1044.62063
[43] Paindaveine, D. (2006). A Chernoff–Savage result for shape. On the nonadmissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206–2220. · Zbl 1101.62045
[44] Puri, M. L. and Sen, P. K. (1985). Nonparametric Methods in General Linear Models . Wiley, New York. · Zbl 0569.62024
[45] Randles, R. H. (1982). On the asymptotic normality of statistics with estimated parameters. Ann. Statist. 10 462–474. · Zbl 0493.62022
[46] Randles, R. H. (1989). A distribution-free multivariate sign test based on interdirections. J. Amer. Statist. Assoc. 84 1045–1050. JSTOR: · Zbl 0702.62039
[47] Schwartz, L. (1973). Théorie des Distributions . Hermann, Paris.
[48] Sugiura, N. (1972). Locally best invariant test for sphericity and the limiting distributions. Ann. Math. Statist. 43 1312–1316. · Zbl 0251.62036
[49] 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. · Zbl 0563.62065
[50] Tyler, D. E. (1982). Radial estimates and the test for sphericity. Biometrika 69 429–436. JSTOR: · Zbl 0501.62041
[51] Tyler, D. E. (1983). Robustness and efficiency properties of scatter matrices. Biometrika 70 411–420. JSTOR: · Zbl 0536.62042
[52] Tyler, D. E. (1987). A distribution-free \(M\)-estimator of multivariate scatter. Ann. Statist. 15 234–251. · Zbl 0628.62053
[53] Tyler, D. E. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika 74 579–589. JSTOR: · Zbl 0628.62054
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