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We propose estimating the scale parameter (mean of the eigenvalues) of thescatter matrix of an unspecified elliptically symmetric distribution usingweights obtained by solving Tyler's M-estimator of the scatter matrix. Theproposed Tyler's weights-based estimate (TWE) of scale is then used toconstruct an affine equivariant Tyler's M-estimator as a weighted samplecovariance matrix using normalized Tyler's weights. We then develop a unifiedframework for estimating the unknown tail parameter of the ellipticaldistribution (such as the degrees of freedom (d.o.f.) $\nu$ of the multivariate$t$ (MVT) distribution). Using the proposed TWE of scale, a new robust estimateof the d.o.f. parameter of MVT distribution is proposed with excellentperformance in heavy-tailed scenarios, outperforming other competing methods.R-package is available that implements the proposed method.

Affine equivariant Tyler's M-estimator applied to tail parameter learning of elliptical distributions