## Semiparametrically efficient rank-based inference for shape. II: Optimal $$R$$-estimation of shape.(English)Zbl 1115.62059

Summary: A class of $$R$$-estimators based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in part I of this paper by M. Hallin and D. Paindaveine [ibid., 2707–2756 (2006; Zbl 1114.62066)] is proposed for the estimation of the shape matrix of an elliptical distribution. These $$R$$-estimators are root-$$n$$ consistent under any radial density $$g$$, without any moment assumptions, and semiparametrically efficient at some prespecified density $$f$$. When based on normal scores, they are uniformly more efficient than the traditional normal-theory estimator based on empirical covariance matrices (the asymptotic normality of which, moreover, requires finite moments of order four), irrespective of the actual underlying elliptical density. They rely on an original rank-based version of L. Le Cam’s one-step methodology [see “Asymptotic methods in statistical decision theory.” (1986; Zbl 0605.62002)] which avoids the unpleasant nonparametnc estimation of cross-information quantities that is generally required in the context of $$R$$-estimation. Although they are not strictly affine-equivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finite-sample performance.

### MSC:

 62H12 Estimation in multivariate analysis 62G20 Asymptotic properties of nonparametric inference 62G10 Nonparametric hypothesis testing 60F05 Central limit and other weak theorems

### Citations:

Zbl 1114.62066; Zbl 0605.62002
Full Text:

### References:

 [1] Adichie, J. N. (1967). Estimates of regression parameters based on rank tests. Ann. Math. Statist. 38 894–904. · Zbl 0152.37102 [2] Antille, A. (1974). A linearized version of the Hodges–Lehmann estimator. Ann. Statist. 2 1308–1313. · Zbl 0296.62035 [3] Bickel, P. J. and Ritov, Y. (1988). Estimating integrated squared density derivatives. Sharp best order of convergence estimates. Sankhyā Ser. A 50 381–393. · Zbl 0676.62037 [4] Cheng, K. F. and Serfling, R. J. (1981). On estimation of a class of efficiency-related parameters. Scand. Actuar. J. 1981 83–92. · Zbl 0457.62034 [5] Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric tests. Ann. Math. Statist. 29 972–994. · Zbl 0092.36501 [6] Draper, D. (1988). Rank-based robust analysis of linear models. I. Exposition and review (with discussion). Statist. Sci. 3 239–271. · Zbl 0955.62606 [7] Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273–1294. · Zbl 0729.62076 [8] Hallin, M., Oja, H. and Paindaveine, D. (2006). Affine-equivariant $$R$$-estimation of shape. Manuscript in preparation. · Zbl 1115.62059 [9] 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 [10] Hallin, M. and Paindaveine, D. (2002). Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence. Bernoulli 8 787–815. · Zbl 1018.62046 [11] Hallin, M. and Paindaveine, D. (2004). Rank-based optimal tests of the adequacy of an elliptic VARMA model. Ann. Statist. 32 2642–2678. · Zbl 1076.62044 [12] Hallin, M. and Paindaveine, D. (2005). Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors. J. Multivariate Anal. 93 122–163. · Zbl 1087.62098 [13] Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707–2756. · Zbl 1114.62066 [14] 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 [15] Hallin, M. and Paindaveine, D. (2006). $$R$$-estimation. Manuscript in preparation. · Zbl 1115.62059 [16] Hallin, M., Vermandele, C. and Werker, B. J. M. (2006). Serial and nonserial sign-and-rank statistics: Asymptotic representation and asymptotic normality. Ann. Statist. 34 254–289. MR2275242 · Zbl 1091.62033 [17] Hallin, M. and Werker, B. J. M. (2003). Semiparametric efficiency, distribution-freeness and invariance. Bernoulli 9 137–165. · Zbl 1020.62042 [18] Hettmansperger, T. P. and Randles, R. (2002). A practical affine equivariant multivariate median. Biometrika 89 851–860. JSTOR: · Zbl 1036.62045 [19] Hodges, J. L. and Lehmann, E. L. (1963). Estimates of location based on rank tests. Ann. Math. Statist. 34 598–611. · Zbl 0203.21105 [20] Jaeckel, L. A. (1972). Estimating regression coefficients by minimizing the dispersion of the residuals. Ann. Math. Statist. 43 1449–1459. · Zbl 0277.62049 [21] John, S. (1971). Some optimal multivariate tests. Biometrika 58 123–127. JSTOR: · Zbl 0218.62055 [22] John, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika 59 169–173. JSTOR: · Zbl 0231.62072 [23] Jurečková, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann. Math. Statist. 40 1889–1900. · Zbl 0188.51003 [24] Jurečková, J. (1971). Nonparametric estimate of regression coefficients. Ann. Math. Statist. 42 1328–1338. · Zbl 0225.62052 [25] Jurečková, J. and Sen, P. K. (1996). Robust Statistical Procedures: Asymptotics and Interrelations . Wiley, New York. · Zbl 0862.62032 [26] Koul, H. (1971). Asymptotic behavior of a class of confidence regions based on ranks in regression. Ann. Math. Statist. 42 466–476. · Zbl 0215.54204 [27] Koul, H. L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models , 2nd ed. Lecture Notes in Statist. 166 . Springer, New York. · Zbl 1007.62047 [28] Koul, H. L., Sievers, G. L. and McKean, J. W. (1987). An estimator of the scale parameter for the rank analysis of linear models under general score functions. Scand. J. Statist. 14 131–141. · Zbl 0628.62035 [29] Kraft, C. H. and van Eeden, C. (1972). Linearized rank estimates and signed-rank estimates for the general linear hypothesis. Ann. Math. Statist. 43 42–57. · Zbl 0238.62045 [30] Kreiss, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112–133. · Zbl 0616.62042 [31] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory . Springer, New York. · Zbl 0605.62002 [32] Lehmann, E. L. (1963). Nonparametric confidence intervals for a shift parameter. Ann. Math. Statist. 34 1507–1512. · Zbl 0136.40407 [33] Lopuhaä, H. P. (1999). Asymptotics of reweighted estimators of multivariate location and scatter. Ann. Statist. 27 1638–1665. · Zbl 0957.62017 [34] Mauchly, J. W. (1940). Significance test for sphericity of a normal $$n$$-variate distribution. Ann. Math. Statist. 11 204–209. · Zbl 0023.24703 [35] 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 [36] Ollila, E., Hettmansperger, T. P. and Oja, H. (2005). Affine equivariant multivariate sign methods. Preprint, Univ. Jyväskylä. · Zbl 1090.62052 [37] Ollila, E., Oja, H. and Croux, C. (2003). The affine equivariant sign covariance matrix: Asymptotic behaviour and efficiencies. J. Multivariate Anal. 87 328–355. · Zbl 1044.62063 [38] 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 [39] Paindaveine, D. (2006). A canonical definition of shape. Submitted for publication. · Zbl 1101.62045 [40] Randles, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. J. Amer. Statist. Assoc. 95 1263–1268. JSTOR: · Zbl 1009.62047 [41] Schweder, T. (1975). Window estimation of the asymptotic variance of rank estimators of location. Scand. J. Statist. 2 113–126. · Zbl 0319.62026 [42] Sen, P. K. (1966). On a distribution-free method of estimating asymptotic efficiency of a class of nonparametric tests. Ann. Math. Statist. 37 1759–1770. · Zbl 0158.37201 [43] Tyler, D. E. (1982). Radial estimates and the test for sphericity. Biometrika 69 429–436. JSTOR: · Zbl 0501.62041 [44] Tyler, D. E. (1983). Robustness and efficiency of scatter matrices. Biometrika 70 411–420. JSTOR: · Zbl 0536.62042 [45] Tyler, D. E. (1987). A distribution-free $$M$$-estimator of multivariate scatter. Ann. Statist. 15 234–251. · Zbl 0628.62053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.