On Robust Estimators of a Sphericity Measure in High Dimension

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