Differentiability of t-functionals of location and scatter

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 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. 1991). It is shown 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 correspondingM-estimators (μn,Σn). In dimension d = 1 only, the tν functional (μ, σ) extends to be defined and weakly continuous at all P .