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

https://doi.org/10.1186/S13660-018-1788-6

Uploaded (2023) | Journal: Journal of Inequalities and Applications

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Abstract

In the present article, by utilizing some inequalities for linearly negative quadrant dependent random variables, we discuss the uniformly asymptotic normality of sample quantiles for linearly negative quadrant dependent samples under mild conditions. The rate of uniform asymptotic normality is presented and the rate of convergence is near O(n-1/4 log n) when the third moment is finite, which extends and improves the corresponding results of Yang et al.

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.

On the Dependence between Functions of Quantile and Dispersion Estimators

arXiv: Statistics Theory, 2019

In this paper, we derive the joint asymptotic distributions of functions of quantile estimators (the non-parametric sample quantile and the parametric location-scale quantile estimator) with functions of measure of dispersion estimators (the sample variance, sample mean absolute deviation, sample median absolute deviation) - assuming an underlying identically and independently distributed sample. We also discuss the conditions required by the use of such estimators. Further, we show that these results can be extended to any higher order absolute central sample moment as measure of dispersion. Aware of the difference in speed of convergence of the two quantile estimators, we compare the impact of the choice of the quantile estimator (and measure of dispersion) on the asymptotic correlations. Then we prove a scaling law for the asymptotic dependence of quantile estimators with measure of dispersion estimators. Finally, we show a good finite sample performance of the asymptotics in sim...

Quantile estimation via distribution fitting

Applicationes Mathematicae, 2019

This paper focuses on nonparametric estimation of quantiles, based on estimators of the distribution function. We review some known and recommended quantile estimators and propose a new one, which has all the desired properties of quantile estimators. The consistency and asymptotic normality of the estimators is proved. The estimators considered are compared in a small simulation study.

High Quantiles of Heavy-Tailed Distributions: Their Estimation

2002

High quantiles of heavy-tailed distributions are estimated under the assumption that the tail is of Pareto type. The distribution of the logarithm of the ratio of the estimate to the true quantile is asymptotically normal. The same is also proved for the Weissman estimate. 1 Let x1 ≤ x2 ≤. .. ≤ xn be the order statistics of a sample. Usually, x (k) , where k = [np] + 1 ([•] denotes the largest integer), is used as an estimate for xp. Since, in general, E[x (k) ] = xp and P {x (k) ≤ x (p) } = 0.5, the quantity x (k) is neither the mean, nor the median for xp. Therefore, estimates are interpolations between order statistics, for example, weighted mean xp = (1 − g)x (j) + gx (j+1) , where j = [np] and g = np − j [1]. 2 This principle can be applied to estimating small quantiles if p is close to 1.

Champernowne transformation in kernel quantile estimation for heavy-tailed distributions

Afrika Statistika, 2011

By transforming a data set with a modification of the Champernowne distribution function, a kernel quantile estimator for heavy-tailed distributions is given. The asymptotic mean squared error (AMSE) of the proposed estimator and related asymptotically optimal bandwidth are evaluated. Some simulations are drawn to show the performance of the obtained results. Résumé. En transformant un ensemble de données avec la fonction de distribution Champernowne modifiée, un estimateurà noyau du quantile pour les distributionsà queues lourdes est donné. L'erreur quadratique moyenne asymptotique (AMSE) de l'estimateur proposé et la fenêtre optimale asymptotique associée sontévaluées. Des simulations sont effectuées pour montrer la performance des résultats obtenus.

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References (20)

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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.

Nonparametric estimation of quantile density function

Computational Statistics & Data Analysis, 2012

In the present article, a new nonparametric estimator of quantile density function is defined and its asymptotic properties are studied. The comparison of the proposed estimator has been made with estimators given by Jones (1992), graphically and in terms of mean square errors for the uncensored and censored cases.

Expansion for moments of regression quantiles with application to nonparametric testing

We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requires the calculation of moments of Bahadur expansions of Nadaraya-Watson quantile regression estimators. This calculation is done by inverting the problem and application of higher order Edgeworth expansions. The moments allow estimation bounds for the accuracy of Bahadur expansions for integrals of kernel quantile estimators. Another application of our method gives asymptotic results for the es...

Expansion for moments of regression quantiles with applications to nonparametric testing

Bernoulli

We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requires the calculation of moments of Nadaraya-Watson quantile regression estimators. This calculation is done by application of higher order Edgeworth expansions.

Semiparametric second-order reduced-bias high quantile estimation

Test, 2009

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.