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.

翻译：本文提出利用泰尔散射矩阵M估计量求解得到的权重，估计未指定椭圆对称分布散射矩阵的尺度参数（特征值均值）。基于此提出的泰尔权值尺度估计量（TWE），通过归一化泰尔权值构造加权样本协方差矩阵，进而构建仿射等变泰尔M估计量。随后我们建立了用于估计椭圆分布未知尾部参数（如多元t分布的自由度ν）的统一框架。利用所提出的TWE尺度估计量，本文提出了一种新的多元t分布自由度参数鲁棒估计方法，在重尾场景下展现出卓越性能，优于其他竞争方法。所提方法已有R语言包可供实现。