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The paper aims at finding widely and smoothly defined nonparametric locationand scatter functionals. As a convenient vehicle, maximum likelihood estimationof the location vector m and scatter matrix S of an elliptically symmetric tdistribution on d-dimensional space with degrees of freedom larger than 1extends to an M-functional defined on all probability distributions P in aweakly open, weakly dense domain U. Here U consists of P not putting too muchmass in hyperplanes of dimension < d, as shown for empirical measures by Kentand 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 distributionfunctions. For k=1,2,..., and other norms, depending on k and more directlyadapted 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 bearbitrarily heavy-tailed. These results imply asymptotic normality of thecorresponding M-estimators (m_n,S_n). In dimension d=1 only, the t functionalsextend to be defined and weakly continuous at all t.

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