Weighted uniform consistency of kernel density estimators
By Evarist Gin'e, Vladimir Koltchinskii, Joel C. Zinn. 2004
48 citations recorded by the source corpus.
Let fn denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψfβ‖∞<∞ for some 0<β<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence ${\sqrt{\frac{nh_{n}^{d}}{2|\log h_{n}^{d}|}}\|\Psi(t)(f_{n}(t)-Ef_{n}(t))\|_{\infty}}$ to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of β is studied where a similar sequence with a different norming converges a.s. either to 0 or to +∞, depending on convergence or divergence of a certain integral involving the tail probabilities of Ψ(X). The results apply as well to some discontinuous not strictly positive densities.
Subfields: Artificial Intelligence, Numerical Analysis, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c9a5fa46c9fe2a0a58581718e3d6dc738f8adebe
How researchers see this paper
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In Artificial Intelligence
- Evarist Gin'e
- Arnak S. Dalalyan
- Armelle Guillou
- Markus Reiß
- Richard Nickl
- David Mason
- Hanie Sedghi
- Philip M. Long
In Numerical Analysis
- Evarist Gin'e
- Markus Reiß
- Arnak S. Dalalyan
- Armelle Guillou
- Richard Nickl
- Yen-Chi Chen
- Sivaraman Balakrishnan
- Vladimir Koltchinskii
In Statistics and Probability
- Richard Nickl
- Evarist Gin'e
- Markus Reiß
- Arnak S. Dalalyan
- Armelle Guillou
- Kolyan Ray
- David Mason
- Kengo Kato
Citations in the corpus
- Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case (2005, background): “In the classical problems of nonparametric estimation, the rates of convergence for the sup-norm loss on a compact set coincide up to a logarithmic factor with the local rates of convergence (Korostelev and Nussbaum [14], Gin´e, Koltchinskii and Zinn [13]).”
- Rates of strong uniform consistency for multivariate kernel density estimators (2002, background): “The hypothesis onK , taken from Giné, Koltchinskii and Zinn [8], is as follows: (K1) K is a bounded, square integrable function in the linear span (the set of finite linear combinations) of functions k 0 satisfying the following property: the subgraph ofk, {(s, u): k(s) u}, can be represented as a f”