Estimation Of Conditional Quantiles Research Articles

Local Polynomial Quantile Regression With Parametric Features

We propose a new approach to conditional quantile function estimation that combines both parametric and nonparametric techniques. At each design point, a global, possibly incorrect, pilot parametric model is locally adjusted through a kernel smoothing fit. The resulting quantile regression estimator behaves like a parametric estimator when the latter is correct and converges to the nonparametric solution as the parametric start deviates from the true underlying model. We give a Bahadur-type representation of the proposed estimator from which consistency and asymptotic normality are derived under an α -mixing assumption. We also propose a practical bandwidth selector based on the plug-in principle and discuss the numerical implementation of the new estimator. Finally, we investigate the performance of the proposed method via simulations and illustrate the methodology with a data example.

Estimation non paramétrique de quantiles conditionnels pour des variables fonctionnelles spatialement dépendantes

Consider Z i = ( X i , Y i ) , i ∈ N N be a F × R -valued measurable strictly stationary spatial process, where F is a semi-metric space. We study a kernel estimator of conditional quantiles of the univariate response variable Y i given the functional variable X i . The main aim of this Note is to prove the almost complete convergence (with rate) of this estimate. To cite this article: A. Laksaci, F. Maref, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Asymptotic properties of conditional quantile estimator for censored dependent observations

In this paper, we establish strong uniform convergence and asymptotic normality of the conditional quantile estimator for the censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The strong uniform convergence in iid framework has recently been discussed by Ould-Said (Stat Probab Lett 76:579–586, 2006). As a by-product, we also obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and censoring distributions under dependence, which is interesting independently.

Mixed-effects models for conditional quantiles with longitudinal data.

We propose a regression method for the estimation of conditional quantiles of a continuous response variable given a set of covariates when the data are dependent. Along with fixed regression coefficients, we introduce random coefficients which we assume to follow a form of multivariate Laplace distribution. In a simulation study, the proposed quantile mixed-effects regression is shown to model the dependence among longitudinal data correctly and estimate the fixed effects efficiently. It performs similarly to the linear mixed model at the central location when the regression errors are symmetrically distributed, but provides more efficient estimates when the errors are over-dispersed. At the same time, it allows the estimation at different locations of conditional distribution, which conveys a comprehensive understanding of data. We illustrate an application to clinical data where the outcome variable of interest is bounded within a closed interval.

A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data

In this paper we study some asymptotic properties of the kernel conditional quantile estimator with randomly left-truncated data which exhibit some kind of dependence. We extend the result obtained by Lemdani, Ould-Saïd and Poulin [16] in the iid case. The uniform strong convergence rate of the estimator under strong mixing hypothesis is obtained.

Nonparametric Estimation of Conditional CDF and Quantile Functions With Mixed Categorical and Continuous Data

We propose a new nonparametric conditional cumulative distribution function kernel estimator that admits a mix of discrete and categorical data along with an associated nonparametric conditional quantile estimator. Bandwidth selection for kernel quantile regression remains an open topic of research. We employ a conditional probability density function-based bandwidth selector proposed by Hall, Racine, and Li that can automatically remove irrelevant variables and has impressive performance in this setting. We provide theoretical underpinnings including rates of convergence and limiting distributions. Simulations demonstrate that this approach performs quite well relative to its peers; two illustrative examples serve to underscore its value in applied settings.

Dynamic quantile models

This paper introduces the Dynamic Additive Quantile (DAQ) model that ensures the monotonicity of conditional quantile estimates. The DAQ model is easily estimable and can be used for computation and updating of the Value-at-Risk. An asymptotically efficient estimator of the DAQ is obtained by maximizing an objective function based on the inverse KLIC measure. An alternative estimator proposed in the paper is the Method of L-Moments estimator (MLM). The MLM estimator is consistent, but generally not fully efficient. Goodness-of-fit tests and diagnostic tools for the assessment of the model are also provided. For illustration, the DAQ model is estimated from a series of returns on the Toronto Stock Exchange (TSX) market index.

Asymptotic properties of a conditional quantile estimator with randomly truncated data

Let Y be a response variable that is subject to left-truncation by a variable T . We consider the problem of estimating its conditional quantile function given a vector of covariates X . We derive almost sure ( a.s.) consistency and asymptotic normality results for a kernel estimate of the conditional quantile function. Simulations are drawn to illustrate the results for finite samples.

Asymptotics for estimation of quantile regressions with truncated infinite-dimensional processes

Many processes can be represented in a simple form as infinite-order linear series. In such cases, an approximate model is often derived as a truncation of the infinite-order process, for estimation on the finite sample. The literature contains a number of asymptotic distributional results for least squares estimation of such finite truncations, but for quantile estimation, results are not available at a level of generality that accommodates time series models used as finite approximations to processes of potentially unbounded order. Here we establish consistency and asymptotic normality for conditional quantile estimation of truncations of such infinite-order linear models, with the truncation order increasing in sample size. We focus on estimation of the model at a given quantile. The proofs use the generalized functions approach and allow for a wide range of time series models as well as other forms of regression model. The results are illustrated with both analytical and simulation examples.

Non-Crossing Non-Parametric Estimates of Quantile Curves

SummarySince the introduction by Koenker and Bassett, quantile regression has become increasingly important in many applications. However, many non-parametric conditional quantile estimates yield crossing quantile curves (calculated for various p ∈ (0, 1)). We propose a new non-parametric estimate of conditional quantiles that avoids this problem. The method uses an initial estimate of the conditional distribution function in the first step and solves the problem of inversion and monotonization with respect to p ∈ (0, 1) simultaneously. It is demonstrated that the new estimates are asymptotically normally distributed with the same asymptotic bias and variance as quantile estimates that are obtained by inversion of a locally constant or locally linear smoothed conditional distribution function. The performance of the new procedure is illustrated by means of a simulation study and some comparisons with the currently available procedures which are similar in spirit with the method proposed are presented.

Kernel Conditional Quantile Estimation for Stationary Processes with Application to Conditional Value-at-Risk

The paper considers kernel estimation of conditional quantiles for both short-range and long-range-dependent processes. Under mild regularity conditions, we obtain Bahadur representations and central limit theorems for kernel quantile estimates of those processes. Our theory is applicable to many price processes of assets in finance. In particular, we present an asymptotic theory for kernel estimates of the value-at-risk (VaR) of the market value of an asset conditional on the historical information or a state process. The results are assessed based on a small simulation and are applied to AT&T monthly returns. Copyright The Author 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

A smooth nonparametric conditional quantile frontier estimator

Traditional estimators for nonparametric frontier models (DEA, FDH) are very sensitive to extreme values/outliers. Recently, Aragon et al. [2005. Nonparametric frontier estimation: a conditional quantile-based approach. Econometric Theory 21, 358–389] proposed a nonparametric α -frontier model and estimator based on a suitably defined conditional quantile which is more robust to extreme values/outliers. Their estimator is based on a nonsmooth empirical conditional distribution. In this paper, we propose a new smooth nonparametric conditional quantile estimator for the α -frontier model. Our estimator is a kernel based conditional quantile estimator that builds on early work of Azzalini [1981. A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68, 326–328]. It is computationally simple, resistant to outliers and extreme values, and smooth. In addition, the estimator is shown to be consistent and n asymptotically normal under mild regularity conditions. We also show that our estimator's variance is smaller than that of the estimator proposed by Aragon et al. A simulation study confirms the asymptotic theory predictions and contrasts our estimator with that of Aragon et al.

Improving Accuracy and Precision of Value-at-Risk Forecasts

As Value-at-Risk (VaR) forecasts are quantile estimates, in this paper we employ a resampling technique alternative to bootstrap, the jackknife method. The delete-d jackknife is specifically designed for non-smooth statistics such as the quantile and produces unbiased statistics because it resamples from the original distribution rather than from the empirical distribution as in the bootstrap. Unlike previous studies that only take into consideration the uncertainty of VaR forecasts arising from the estimation of conditional volatilities of returns, we also account for the uncertainty resulted from the estimation of the conditional quantiles of the simulated return series by using the median unbiased quantile estimator. Applied to five proxy portfolios for managed funds in Australia, the proposed technique is shown to have modest improvement over the other eight benchmark models in terms of statistical and regulatory tests, but it provides much more accurate VaR forecasts based on statistical loss measures. In particular, considerable improvement is achieved in forecast precision.

Multivariate wavelet-based shape-preserving estimation for dependent observations

We introduce a new approach to shape-preserving estimation of cumulative distribution functions and probability density functions using the wavelet methodology for multivariate dependent data. Our estimators preserve shape constraints such as monotonicity, positivity and integration to one, and allow for low spatial regularity of the underlying functions. We discuss conditional quantile estimation for financial time series data as an application. Our methodology can be implemented with B-splines. We show by means of Monte Carlo simulations that it performs well in finite samples and for a data-driven choice of the resolution level.

Conditional quantile estimation by local logistic regression†

In this article, we propose a new nonparametric estimator of the conditional quantile function. It is based on locally fitting a logistic model. We compare the new proposal with some existing methods. Those include the double-kernel technique of Yu and Jones (1998), the adjusted version of the Nadaraya–Watson estimator suggested by Hall et al. (1999) and the approach by Koenker and Bassett (1978) based on the ‘check function’ loss. The comparison is done by asymptotic mean squared error and a simulation study. The four estimators have the same asymptotic variance, but their first-order biases are different. We also propose a new automatic smoothing parameter selection method in quantile estimation. We analyze the finite sample properties of the quantile estimators using the proposed bandwidth selectors. We find that the new method outperforms the others in most cases of the numerical experiments.

On the uth geometric conditional quantile

Motivated by Chaudhuri's work [1996. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91, 862–872] on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.

On the u-th Geometric Conditional Quantile

Motivated by Chaudhuri's work (1996) on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions in high dimensional spaces. We establish a Bahadur type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.

On probabilistic properties of conditional medians and quantiles

Conditional medians and quantiles are frequently used in analyzing time series data with heavy tails for their robustness properties. Most of the available literature focusses on statistical estimation of conditional quantiles and large sample behavior of the proposed estimators, whereas probabilistic properties of the true conditional medians themselves have not been fully explored (Tomkins [1975. On conditional medians. Ann. Probab. 3, 375–379; 1978. Convergence properties of conditional medians. Canad. J. Statist. 6, 169–177]). It is well-known that not all properties of conditional expectations have analogues for conditional medians. In this short note, we study to what extent analogues of certain properties of conditional expectations hold for conditional medians and generalize some of these properties to any general conditional quantile.

Regression quantiles under dependent censoring

In a recent paper, Braekers and Veraverbeke [Braekers, R. and Veraverbeke, N., 2005, A copula-graphic estimator for the conditional survival function under dependent censoring. Canadian Journal of Statistics, 33, 429–447.] obtained asymptotic results for an estimator of the conditional distribution function of the lifetime, in a fixed design regression model with dependence between lifetime and censoring time. The author complements these findings by corresponding results for the conditional quantile estimator. The main results are uniform strong consistency and a Bahadur-type representation. The latter is an invaluable tool for deriving further asymptotic results, such as the weak convergence of the quantile process.

A strong uniform convergence rate of kernel conditional quantile estimator under random censorship

In this paper, we study the kernel conditional quantile estimation for censored data and a uniform strong convergence rate of the resulting estimator is established. The rate obtained here is the same as that for uncensored case.