Semiparametrically efficient rank-based inference for shape II. Optimal R-estimation of shape

Annals of Statistics

Volumn 34, Issue 6, 2006, Pages 2757-2789

  Hallin, Marc ^a   Oja, Hannu ^b,c   Paindaveine, Davy ^a  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

^b TAMPERE UNIVERSITY OF TECHNOLOGY   (Finland)

^c UNIVERSITY OF TAMPERE   (Finland)

Author keywords

Affine equivariance; Elliptical densities; Local asymptotic normality; Multivariate ranks and signs; One step estimation; R estimation; Sermiparametric efficiency; Shape matrix

Indexed keywords

EID: 34547337738     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000948     Document Type: Article

References (45)

1. ADICHIE, J. N. (1967). Estimates of regression parameters based on rank tests. Ann. Math. Statist. 38 894-904. MR0247715
2. ANTILLE, A. (1974). A linearized version of the Hodges-Lehmann estimator. Ann. Statist. 2 1308-1313. MR0365868
3. BICKEL, P. J. and RITOV, Y. (1988). Estimating integrated squared density derivatives. Sharp best order of convergence estimates. Sankhya Ser. A 50 381-393. MR1065550
4. CHENG, K. F. and SERFLING, R. J. (1981). On estimation of a class of efficiency-related parameters. Scand. Actuar. J. 1981 83-92. MR0623403
5. CHERNOFF, H. and SAVAGE, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric tests. Ann. Math. Statist. 29 972-994. MR0100322
6. DRAPER, D. (1988). Rank-based robust analysis of linear models. I. Exposition and review (with discussion). Statist. Sci. 3 239-271. MR0968391
7. FAN, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273-1294. MR1126325
8. HALLIN, M., OJA, H. and PAINDAVEINE, D. (2006). Affine-equivariant K-estimation of shape. Manuscript in preparation.
9. HALLIN, M. and PAINDAVEINE, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103-1133. MR1926170
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. MR1963662
11. HALLIN, M. and PAINDAVEINE, D. (2004). Rank-based optimal tests of the adequacy of an elliptic VARMA model. Ann. Statist. 32 2642-2678. MR2153998
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. MR2119768
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.
14. HALLIN, M. and PAINDAVEINE, D. (2006). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 327-350.
15. HALLIN, M. and PAINDAVEINE, D. (2006). R-estimation. Manuscript in preparation.
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
17. HALLIN, M. and WERKER, B. J. M. (2003). Semiparametric efficiency, distribution-freeness and invariance. Bernoulli 9 137-165. MR1963675
18. HETTMANSPERGER, T. P. and RANDLES, R. (2002). A practical affine equivariant multivariate median. Biometrika 89 851-860. MR1946515
19. HODGES, J. L. and LEHMANN, E. L. (1963). Estimates of location based on rank tests. Ann. Math. Statist. 34 598-611. MR0152070
20. JAECKEL, L. A. (1972). Estimating regression coefficients by minimizing the dispersion of the residuals. Ann. Math. Statist. 43 1449-1459. MR0348930
21. JOHN, S. (1971). Some optimal multivariate tests. Biometrika 58 123-127. MR0275568
22. JOHN, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika 59 169-173. MR0312619
23. JUREČKOVÁ, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann. Math. Statist. 40 1889-1900. MR0248931
24. JUREČKOVÁ, J. (1971). Nonparametric estimate of regression coefficients. Ann. Math. Statist. 421328-1338. MR0295487
25. JUREČKOVÁ, J. and SEN, P. K. (1996). Robust Statistical Procedures: Asymptotics and Interrelations. Wiley, New York. MR1387346
26. KOUL, H. (1971). Asymptotic behavior of a class of confidence regions based on ranks in regression. Ann. Math. Statist. 42 466-476. MR0288896
27. KOUL, H. L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models, 2nd ed. Lecture Notes in Statist. 166. Springer, New York. MR1911855
28. KOUL, H. L., SIE VERS, G. L. and MCK EAN, 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. MR0913258
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. MR0301857
30. KREISS, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112-133. MR0885727
31. LE CAM, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York. MR0856411
32. LEHMANN, E. L. (1963). Nonparametric confidence intervals for a shift parameter. Ann. Math. Statist. 34 1507-1512. MR0164412
33. LOPUHAÄ, H. P. (1999). Asymptotics of reweighted estimators of multivariate location and scatter. Ann. Statist. 27 1638-1665. MR1742503
34. MAUCHLY, J. W. (1940). Significance test for sphericity of a normal n-variate distribution. Ann. Math. Statist. 11 204-209. MR0002084
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. MR0570502
36. OLLILA, E., HETTMANSPERGER, T. P. and OJA, H. (2005). Affine equivariant multivariate sign methods. Preprint, Univ. Jyväskylä.
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. MR2016942
38. PAINDAVEINE, D. (2006). A Chernoff-Savage result for shape. On the nonadmissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206-2220.
39. PAINDAVEINE, D. (2006). A canonical definition of shape. Submitted for publication.
40. RANDLES, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. J. Amer. Statist. Assoc. 95 1263-1268. MR1792189
41. SCHWEDER, T. (1975). Window estimation of the asymptotic variance of rank estimators of location. Scand. J. Statist. 2 113-126. MR0411042
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. MR0199928
43. TYLER, D. E. (1982). Radial estimates and the test for sphericity. Biometrika 69 429-436. MR0671982
44. TYLER, D. E. (1983). Robustness and efficiency of scatter matrices. Biometrika 70 411-420. MR0712028
45. TYLER, D. E. (1987). A distribution-free M-estimator of multivariate scatter. Ann. Statist. 15 234-251. MR0885734