Differentiability of M-functionals of location and scatter based on t likelihoods

TL;DR

This paper establishes the smoothness and differentiability properties of M-functionals of location and scatter based on t likelihoods, enabling robust statistical inference for heavy-tailed distributions in multivariate settings.

Contribution

It proves the analytic and differentiability properties of these M-functionals on a broad class of probability distributions, extending their applicability and enabling asymptotic analysis.

Findings

M-functionals are analytic on a large domain of probability measures.

They are continuously differentiable of any order, facilitating the delta-method.

The associated M-estimators are asymptotically normal.

Abstract

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector m and scatter matrix S of an elliptically symmetric t distribution on d-dimensional space with degrees of freedom larger than 1 extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P not putting too much mass in hyperplanes of dimension < d, as shown for empirical measures by Kent and Tyler, Ann. Statist. 1991. It is shown here that (m,S) is analytic on U, for the bounded Lipschitz norm, or for d=1, for the sup norm on distribution functions. For k=1,2,..., and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (m,S) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (m_n,S_n). In dimension d=1 only, the t functionals extend to be defined and weakly continuous at all t.