Differentiability of t-functionals of location and scatter

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 μ and scatter matrix Σ of an elliptically symmetric t distribution on ℝ^d with degrees of freedom ν>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 putting not too much mass in hyperplanes of dimension <d, as shown for empirical measures by Kent and Tyler [Ann. Statist. 19 (1991) 2102–2119]. It will be seen here that (μ, Σ) 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 (μ, Σ) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (μ_n, Σ_n). In dimension d=1 only, the t_ν functional (μ, σ) extends to be defined and weakly continuous at all P.