Robust $M$-Estimators of Multivariate Location and Scatter

Abstract

Let $\mathbf{x}_1,\cdots, \mathbf{x}_n$ be a sample from an $m$-variate distribution which is spherically symmetric up to an affine transformation. This paper deals with the robust estimation of the location vector $\mathbf{t}$ and scatter matrix $\mathbf{V}$ by means of "$M$-estimators," defined as solutions of the system: $\sum_i u_1(d_i)(\mathbf{x}_i - \mathbf{t}) = \mathbf{0}$ and $n^{-1}\sum_i u_2(d_i^2)(\mathbf{x}_i - \mathbf{t})(\mathbf{x}_i - \mathbf{t})' = \mathbf{V}$, where $d_i^2 = (\mathbf{x}_i - \mathbf{t})'\mathbf{V}^{-1}(\mathbf{x}_i - \mathbf{t})$. Existence and uniqueness of solutions of this system are proved under general assumptions about the functions $u_1$ and $u_2$. Then the estimators are shown to be consistent and asymptotically normal. The breakdown bound and the influence function are calculated, showing some weaknesses of the estimates for high dimensionality. An algorithm for the numerical calculation of the estimators is described. Finally, numerical values of asymptotic variances, and Monte Carlo small-sample results are exhibited.