This article is cited in 3 papers

Central limit theorem of the perturbed sample quantile for a sequence of $m$-dependent nonstationary random process

Shan Sun

Dept. of Mathematics, Indiana University, Indiana, USA

Abstract: 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 $\hat\xi_{np}=\inf\{x\in\mathbf{R},\widehat F_n(x)\ge p\}$, $0<p<1$, 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.

Received: 29.08.1991

Language: English

 English version: Theory of Probability and its Applications, 1995, 40:1, 116–129

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