\(k\)-step shape estimators based on spatial signs and ranks. (English) Zbl 1372.62007

Summary: The shape matrix estimators based on spatial sign and rank vectors are considered. The estimators considered here are slight modifications of the estimators introduced in [L. Dümbgen, Ann. Inst. Stat. Math. 50, No. 3, 471–491 (1998; Zbl 0912.62061)] and [H. Oja and [O. H. Randles, Stat. Sci. 19, No. 4, 598–605 (2004; Zbl 1100.62567)] and further studied for example in [S. Sirkiä et al., J. Nonparametric Stat. 21, No. 2, 155–176 (2009; Zbl 1359.62150)]. The shape estimators are computed using pairwise differences of the observed data, therefore there is no need to estimate the location center of the data. When the estimator is based on signs, the use of differences also implies that the estimators have the so called independence property if the estimator, that is used as an initial estimator, has it. The influence functions and limiting distributions of the estimators are derived at the multivariate elliptical case. The estimators are shown to be highly efficient in the multinormal case, and for heavy-tailed distributions they outperform the shape estimator based on sample covariance matrix.


62G05 Nonparametric estimation
62H12 Estimation in multivariate analysis


Full Text: DOI


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