## Abstract

The need to test (or estimate) sphericity arises in various applications

in statistics, and thus the problem has been investigated in numerous papers.

Recently, estimates of a sphericity measure are needed in high-dimensional shrink-

age covariance matrix estimation problems, wherein the (oracle) shrinkage param-

eter minimizing the mean squared error (MSE) depends on the unknown sphericity

parameter. The purpose of this chapter is to investigate the performance of robust

sphericity measure estimators recently proposed within the framework of elliptically

symmetric distributions when the data dimensionality, p, is of similar magnitude as

the sample size, n. The population measure of sphericity that we consider here is

defined as the ratio of the mean of the squared eigenvalues of the scatter matrix

parameter relative to the mean of its eigenvalues squared. We illustrate that robust

sphericity estimators based on the spatial sign covariance matrix (SSCM) or M-

estimators of scatter matrix provide superior performance for diverse covariance

matrix models compared to sphericity estimators based on the sample covariance

matrix (SCM) when distributions are heavy-tailed and .n = O(p). At the same time,

they provide equivalent performance when the data are Gaussian. Our examples also

illustrate the important role that the sphericity plays in determining the attainable

accuracy of the SCM

in statistics, and thus the problem has been investigated in numerous papers.

Recently, estimates of a sphericity measure are needed in high-dimensional shrink-

age covariance matrix estimation problems, wherein the (oracle) shrinkage param-

eter minimizing the mean squared error (MSE) depends on the unknown sphericity

parameter. The purpose of this chapter is to investigate the performance of robust

sphericity measure estimators recently proposed within the framework of elliptically

symmetric distributions when the data dimensionality, p, is of similar magnitude as

the sample size, n. The population measure of sphericity that we consider here is

defined as the ratio of the mean of the squared eigenvalues of the scatter matrix

parameter relative to the mean of its eigenvalues squared. We illustrate that robust

sphericity estimators based on the spatial sign covariance matrix (SSCM) or M-

estimators of scatter matrix provide superior performance for diverse covariance

matrix models compared to sphericity estimators based on the sample covariance

matrix (SCM) when distributions are heavy-tailed and .n = O(p). At the same time,

they provide equivalent performance when the data are Gaussian. Our examples also

illustrate the important role that the sphericity plays in determining the attainable

accuracy of the SCM

Original language | English |
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Title of host publication | Robust and Multivariate Statistical Methods |

Subtitle of host publication | Festschrift in Honor of David E. Tyler |

Editors | Mengxi Yi, Klaus Nordhausen |

Place of Publication | Cham |

Publisher | Springer |

Pages | 179-195 |

ISBN (Electronic) | 978-3-031-22687-8 |

ISBN (Print) | 978-3-031-22686-1 |

DOIs | |

Publication status | Published - 2023 |

MoE publication type | A3 Book section, Chapters in research books |