On the estimation of the norm of the n-linear symmetric operation

Kopeć, J.; Musielak, Julian

doi:10.4064/sm-15-1-29-30

Cited by 5 publications

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“…Chae (1985), Theorem 4.13. The bounds are sharp in general Banach spaces [Kopeć and Musielak (1956)] but if X is a Hilbert space we have g T ≡ T [Bochnak and Siciak (1971)].…”

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confidence: 99%

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

Dudley,

Sidenko,

Wang

2008

Preprint

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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. 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 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 .

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“…Chae (1985), Theorem 4.13. The bounds are sharp in general Banach spaces [Kopeć and Musielak (1956)] but if X is a Hilbert space we have g T ≡ T [Bochnak and Siciak (1971)].…”

mentioning

confidence: 99%