  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.) $\nu$ 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.
