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Differentiability of t-functionals of location and scatter

Author(s)
Dudley, Richard M.; Sidenko, Sergiy; Wang, Zuoqin
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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 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.
Date issued
2011-06
URI
http://hdl.handle.net/1721.1/63108
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Annals of Statistics
Publisher
Institute of Mathematical Statistics
Citation
Dudley, R. M., Sergiy Sidenko and Zuoqin Wang. "Differentiability of t-functionals of location and scatter." Ann. Statist. 37.2 (2009): 939-960.
Version: Author's final manuscript
ISSN
0090-5364

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