### Abstract

Original language | English |
---|---|

Pages (from-to) | 54-67 |

Number of pages | 14 |

Journal | Journal of Multivariate Analysis |

Volume | 144 |

Early online date | 10 Nov 2015 |

DOIs | |

Publication status | Published - Feb 2016 |

### Fingerprint

### Keywords

- stat.ME
- math.ST
- stat.TH
- 62H12, 62G05, 62G35
- scale estimator
- spatial sign correlation
- spatial sign covariance matrix

### Cite this

**Asymptotics of the two-stage spatial sign correlation.** / Dürre, Alexander; Vogel, Daniel.

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 144, pp. 54-67. https://doi.org/10.1016/j.jmva.2015.10.011

}

TY - JOUR

T1 - Asymptotics of the two-stage spatial sign correlation

AU - Dürre, Alexander

AU - Vogel, Daniel

N1 - Acknowledgments This research was supported in part by the Collaborative Research Grant 823 of the German Research Foundation. The authors wish to thank the editors and referees for their careful handling of the manuscript. They further acknowledge the anonymous referees of the article Spatial sign correlation (J. Multivariate Anal. 135, pages 89–105, 2015), who independently of each other suggested to further explore the properties of two-stage spatial sign correlation.

PY - 2016/2

Y1 - 2016/2

N2 - The spatial sign correlation (Dürre et al., 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. Dürre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation in the same vein a Fisher’s z-transform. This variance-stabilizing transform is valid for all elliptical distributions and yields very accurate confidence intervals.

AB - The spatial sign correlation (Dürre et al., 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. Dürre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation in the same vein a Fisher’s z-transform. This variance-stabilizing transform is valid for all elliptical distributions and yields very accurate confidence intervals.

KW - stat.ME

KW - math.ST

KW - stat.TH

KW - 62H12, 62G05, 62G35

KW - scale estimator

KW - spatial sign correlation

KW - spatial sign covariance matrix

U2 - 10.1016/j.jmva.2015.10.011

DO - 10.1016/j.jmva.2015.10.011

M3 - Article

VL - 144

SP - 54

EP - 67

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

ER -