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April 2009 Differentiability of t-functionals of location and scatter
R. M. Dudley, Sergiy Sidenko, Zuoqin Wang
Ann. Statist. 37(2): 939-960 (April 2009). DOI: 10.1214/08-AOS592


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 ℝd 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 (μn, Σn). In dimension d=1 only, the tν functional (μ, σ) extends to be defined and weakly continuous at all P.


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R. M. Dudley. Sergiy Sidenko. Zuoqin Wang. "Differentiability of t-functionals of location and scatter." Ann. Statist. 37 (2) 939 - 960, April 2009.


Published: April 2009
First available in Project Euclid: 10 March 2009

zbMATH: 1162.62023
MathSciNet: MR2502656
Digital Object Identifier: 10.1214/08-AOS592

Primary: 62G05 , 62GH20
Secondary: 62G35

Keywords: Affinely equivariant , Fréchet differentiable , weakly continuous

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 2 • April 2009
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