Central limit theorem of the perturbed sample quantile for a~sequence of $m$-dependent nonstationary random process
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 143-158
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Given a sequence $X_i$, $i\ge1$, of $m$-dependent nonstationary random variables, the usual perturbed empirical distribution function is $\widehat F_n(x)=n^{-1}\sum_{i=1}^nK_n(x-X_i)$, where $K_n$, $n\ge1$, is a sequence of continuous distribution functions converging weakly to the distribution function with a unit mass at zero. In this paper, we study the perturbed sample quantile estimator $\hat\xi_{np}=\inf\{x\in\mathbf{R},\widehat F_n(x)\ge p\}$, $0$, based on a kernel $k$ associated with $K_n$ and a sequence of window-widths $a_n>0$. Under suitable assumptions, we prove the weak as well as the strong consistency of $\hat\xi_{np}$ and also provide sufficient conditions for the asymptotic normality of $\hat\xi_{np}$. Our central limit theorem for $\hat\xi_{np}$ generalizes a result of Sen [15] and also extends the results of Nadarya [8] and Ralescu and Sun [12].
Keywords:
perturbed sample quantile, central limit theorem, $m$-dependent nonstationary random variables, weak and strong consistency, perturbed empirical distribution functions.
@article{TVP_1995_40_1_a7,
author = {Shan Sun},
title = {Central limit theorem of the perturbed sample quantile for a~sequence of $m$-dependent nonstationary random process},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {143--158},
publisher = {mathdoc},
volume = {40},
number = {1},
year = {1995},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a7/}
}
TY - JOUR AU - Shan Sun TI - Central limit theorem of the perturbed sample quantile for a~sequence of $m$-dependent nonstationary random process JO - Teoriâ veroâtnostej i ee primeneniâ PY - 1995 SP - 143 EP - 158 VL - 40 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a7/ LA - en ID - TVP_1995_40_1_a7 ER -
%0 Journal Article %A Shan Sun %T Central limit theorem of the perturbed sample quantile for a~sequence of $m$-dependent nonstationary random process %J Teoriâ veroâtnostej i ee primeneniâ %D 1995 %P 143-158 %V 40 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a7/ %G en %F TVP_1995_40_1_a7
Shan Sun. Central limit theorem of the perturbed sample quantile for a~sequence of $m$-dependent nonstationary random process. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 143-158. http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a7/