Asymptotic theorems for kernel U-quantiles
type: Item
instance of: scholarly article
title: Asymptotic theorems for kernel U-quantiles
zbMATH Open document ID: 1276.62028
DOI: 10.1214/12-EJS687
author: Stefan S. Ralescu
published in: Electronic Journal of Statistics
publication date: 2013-05-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ejs/1334754010
review text: Let \(X_1, X_2, \ldots\) be independent and identically distributed random variables and let \(h(x_1, \ldots, x_m)\) a real-valued function that is symmetric in its arguments. Let \[ H(y) = P(h(X_1, \ldots, X_n) \leq y)\;\text{and}\;H_n(y) = \sum I(h(X_{i_1},\ldots, X_{i_m}) \leq y)/n_{(m)}, \] where the sum is over the \(n_{(m)} = n(n-1)\ldots(n-m)\) \(m\)-tuples \((i_1, \ldots,i_m)\) of distinct elements drawn from \(\{1,2, \ldots, n\}\). A natural estimator for the parameter \(\xi_p = \inf \{x: H(x)\geq p\}\) is the \(U\)-quantile \(H_n^{-1}(p)\). This paper studies alternative smoothed estimators for \(\xi_p\) which are kernel \(U\)-quantiles. Sufficient condtions are given to establish a strong Bahadur representation theorem for the kernel \(U\)-quantile and an invariance principle for the estimator. The asymptotic normality for estimators based on randomly indexed sequences of random variables is also established.
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 6166972
zbMATH Keywords: Bahadur representation
zbMATH Keywords: kernel estimator
zbMATH Keywords: invariance principles
reviewed by: Neville Weber
MaRDI profile type: Publication
cites work: The Bahadur-Kiefer representation for \(U\)-quantiles
cites work: Generalized order statistics, Bahadur representations, and sequential nonparametric fixed-width confidence intervals
cites work: The almost sure invariance principle for the empirical process of U- statistic structure
cites work: Relative deficiency of kernel type estimators of quantiles
cites work: Asymptotic normality of the kernel quantile estimator
cites work: Weak Convergence of $U$-Statistics and Von Mises' Differentiable Statistical Functions
cites work: Nonparametric Statistical Data Modeling
cites work: Q4030906
cites work: Strong approximation theorems for integrated kernel quantiles
cites work: Generalized L-, M-, and R-statistics
cites work: Kernel Quantile Estimators
cites work: A kernel-type estimator for generalized quantiles
cites work: Bahadur representation for \(U\)-quantiles of dependent data
cites work: A Smooth Nonparametric Estimator of a Quantile Function
description: scientific article; zbMATH DE number 6166972