Uniformly asymptotic normality of sample quantiles estimator for linearly negative quadrant dependent samples (original) (raw)

On Asymptotic Behavior of Weighted Sample Quantiles

Mathematical Methods of Statistics

We study the limiting behavior of L-statistics with the “delta type” weight sequences, of which the O-statistics and the usual kernel quantile estimators are special cases. Deviations of these statistics from the sample quantiles are shown to have (after proper normalization) Gaussian limit distributions. We evaluate the main term of the asymptotics of the covariance of this deviation with the corresponding sample quantile. This gives also the asymptotic relative deficiency of sample quantiles w.r.t. our L-statistics.

On M -estimators and normal quantiles

The Annals of Statistics, 2003

This paper explores a class of robust estimators of normal quantiles filling the gap between maximum likelihood estimators and empirical quantiles. Our estimators are linear combinations of M-estimators. Their asymptotic variances can be arbitrarily close to variances of the maximum likelihood estimators. Compared with empirical quantiles, the new estimators offer considerable reduction of variance at near normal probability distributions.

Berry-Esséen rate in asymptotic normality for perturbed sample quantiles

Metrika, 1994

In estimating quantiles with a sample of size N obtained from a distribution F, the perturbed sample quantiles based on a kernel function k have been investigated by many authors. It is well known that their behaviour depends on the choices of "window-width", say w N. Under suitable and reasonably mild assumptions on F and k, Ralescu and Sun (1993) have recently proven that lim N ~ N ~/4w N = 0 is the necessary and sufficient condition for the asymptotic normality of the perturbed sample quantiles. In this paper, their rate of convergence is investigated. It turns out that the optimal Berry-Ess6en rate of O(N-~/2) can be achieved by choosing the window-width suitably, say w N = O(N-~/2). The obtained results, in addition to being explicit enough to verify the sufficient condition for the asymptotic normality, improve Ralescu's (1992) result of which the rate is of order (log N)N-U2.

On M-estimators of approximate quantiles and approximate conditional quantiles

M-estimators introduced in Huber (1964) provide a class of robust estimators of a center of symmetry of a symmetric probability distribution which also have very high efficiency at the model. However it is not clear what they do estimate when the probability distributions are nonsymmetric. In this paper we first show that in the case of arbitrary, not necessarily symmetric probabilty distributions, some M-functionals coincide with quantiles of a convolution of the parent distribution and a probability distribution determined by the M-function. Next, we consider linear combinations of two such M-functionals. These combinations reduce the contribution coming from the convoluting factor and related to the use of the M-functionals and provide good approximation of normal quantiles. We show that use of particular M-functions, which we call probit M-functions, results in recovering exactly the quantiles of the normal probability distribution. Our estimators of quantiles are the empirical ...

On Quantile‐based Asymmetric Family of Distributions: Properties and Inference

International Statistical Review, 2019

SummaryIn this paper, we provide a detailed study of a general family of asymmetric densities. In the general framework, we establish expressions for important characteristics of the distributions and discuss estimation of the parameters via method‐of‐moments as well as maximum likelihood estimation. Asymptotic normality results for the estimators are provided. The results under the general framework are then applied to some specific examples of asymmetric densities. The use of the asymmetric densities is illustrated in a real‐data analysis.

A Berry-Esseen theorem for sample quantiles under association

Communications in Statistics - Theory and Methods

In this paper, the uniformly asymptotic normality for sample quantiles of associated random variables is investigated under some conditions on the decay of the covariances. We obtain the rate of normal approximation of order O(n −1/2 log 2 n) if the covariances decrease exponentially to 0. The best rate is shown as O(n −1/3) under a polynomial decay of the covariances.