Recursive kernel density estimators under a weak dependence condition |
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Authors: | Lanh Tat Tran |
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Institution: | (1) Department of Mathematics, Indiana University, 47405 Bloomington, IN, U.S.A. |
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Abstract: | Let X
t, t= ..., \s-1,0,1,... be a strietly stationary sequence of random variables (r.v.'s) defined on a probability space ( ,P) and taking values in R
d.Let X
1,...,X
nbe n consecutive observations of X
t.Let f be the density of X
1.As an estimator of f(x), we shall consider % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqGaaO% qaaiqadAgagaqcamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGa% aiykaiabg2da9iaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcda% aeWbqaaiaadkgadaWgaaWcbaGaamOAaaqabaGcdaahaaWcbeqaaiab% gkHiTiaadsgaaaGccaWGlbGaaiikaiaacIcacaWG4bGaeyOeI0Iaam% iwamaaBaaaleaacaWGQbaabeaakiaacMcacaGGVaGaamOyamaaBaaa% leaacaWGQbaabeaakiaacMcaaSqaaiaadQgacqGH9aqpcaaIXaaaba% GaamOBaaqdcqGHris5aaaa!58A9!\\hat f_n (x) = n^{ - 1} \sum\limits_{j = 1}^n {b_j ^{ - d} K((x - X_j )/b_j )} \]. Here K is a kernel function and b
nis a esquence of bandwidths tending to zero as n . The asymptotic distribution and uniform convergence of f
n are obtained under general conditions. Appropriate bandwidths are given explicitly. The process X
tis assumed to satisfy a weak dependence condition defined in terms of joint densities. The results are applicable to a large class of time series models. |
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Keywords: | Asymptotic normality uniform convergence absolute regularity density estimation kernel bandwidth |
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