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This article is cited in 1 scientific paper (total in 1 paper)
Probability theory and mathematical statistics
About conditions of gaussian approximation of kernel estimates for distribution density
A. S. Kartashova, A. I. Sakhanenkob a Novosibirsk State University, st. Pirogova, 2, 630090, Novosibirsk, Russia
b Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
Abstract:
Recently E. Gine, V. Koltchinskii and L. Sakhanenko (2004) investigated asymptotical behavior of a random variable of the form $\sqrt{n h_n} \sup\nolimits_{t \in \mathbf{R}} \left | \psi(t) (f_n(t) - \mathbf{E} f_n (t)) \right | $ with some weight function $\psi(t)$, where $f_n$ is a kernel density estimator. The proof of their limit theorems consists of a large number of technically difficult stages and uses more than ten bulky assumptions. In this work we show that under simpler and wider conditions the above stated problem is reduced to the study of asymptotics of a supremum of some special Gaussian process. The obtained result can be used in further investigation of functionals based on empirical processes and kernel density estimators. Our proof is based on the well-known approximation of Komlos, Major and Tusnady (1975).
Keywords:
kernel density estimators, brownian motion, brownian bridge, KMT approximation, function of bounded variation.
Received August 29, 2015, published November 5, 2015
Citation:
A. S. Kartashov, A. I. Sakhanenko, “About conditions of gaussian approximation of kernel estimates for distribution density”, Sib. Èlektron. Mat. Izv., 12 (2015), 766–776
Linking options:
https://www.mathnet.ru/eng/semr625 https://www.mathnet.ru/eng/semr/v12/p766
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