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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
Аннотация:
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].
Ключевые слова:
perturbed sample quantile, central limit theorem,
$m$-dependent nonstationary random variables, weak and strong consistency, perturbed empirical distribution functions.
Поступила в редакцию: 29.08.1991
Язык публикации: английский
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