Affine Equivariant Tyler's M-Estimator Applied to Tail Parameter Learning of Elliptical Distributions

Abstract

We propose estimating the scale parameter (mean of the eigenvalues) of the scatter matrix of an unspecified elliptically symmetric distribution using weights obtained by solving Tyler's M-estimator of the scatter matrix. The proposed Tyler's weights-based estimate (TWE) of scale is then used to construct an affine equivariant Tyler's M-estimator as a weighted sample covariance matrix using normalized Tyler's weights. We then develop a unified framework for estimating the unknown tail parameter of the elliptical distribution (such as the degrees of freedom (d.o.f.) ν of the multivariate t (MVT) distribution). Using the proposed TWE of scale, a new robust estimate of the d.o.f. parameter of MVT distribution is proposed with excellent performance in heavy-tailed scenarios, outperforming other competing methods. R-package is available that implements the proposed method.