We study the bias of classical quantile regression and instrumental variable quantile regression estimators. While being asymptotically first-order unbiased, these estimators can have non-negligible second-order biases. We derive a higher-order stochastic expansion of these estimators using empirical process theory. Based on this expansion, we derive an explicit formula for the second-order bias and propose a feasible bias correction procedure that uses finite-difference estimators of the bias components. The proposed bias correction method performs well in simulations. We provide an empirical illustration using Engel's classical data on household food expenditure.
Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically, it obtains the large sample properties of extremal (extreme order and intermediate order) quantile regression estimators for the linear quantile regression model with the tails restricted to the domain of minimum attraction and closed under tail equivalence a…
Quantile regression predicts the $τ$-quantile of the conditional distribution of a response variable given the explanatory variable for $τ\in(0,1)$. The aim of this paper is to establish the asymptotic distribution of the quantile estimator obtained by penalized spline method. A simulation and an exploration of real data are performed to validate our results.
This work proposes new inference methods for a regression coefficient of interest in a (heterogeneous) quantile regression model. We consider a high-dimensional model where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable approximation to the conditional quantile function. The proposed methods are (explicitly or implicitly) based on orthogonal score functions that protect against moderate model selection mistakes, which are often inevitable in the approximately sparse model considered in the present paper. We establish the unif…
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. …
We address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate la…
In several different fields, there is interest in analyzing the upper or lower tail quantile of the underlying distribution rather than mean or center quantile. However, the investigation of the tail quantile is difficult because of data sparsity. In this paper, we attempt to develop nonparametric quantile regression for the extremal quantile level. In extremal quantile regression, there are two types of technical conditions of the order of convergence of the quantile level: intermediate order or extreme order. For the intermediate order quantile, the ordinary nonparametric estimator is used. …
We address the estimation of "extreme" conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order.
Conditional quantiles provide a natural tool for reporting results from regression analyses based on semiparametric transformation models. We consider their estimation and construction of confidence sets in the presence of censoring.
Nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data. In a large class of such models (including probit, proportional hazard and quantile regression to name just a few) it is impossible to difference out individual effects, and inference is usually justified in a `large n large T' asymptotic framework. However, there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions (such as probit, and proportional hazard) and quantile regression. In the present paper we …
For second order stationary processes, the spectral distribution function is uniquely deter- mined by the autocovariance functions of the processes. We define the quantiles of the spectral distribution function and propose two estimators for the quantiles. Asymptotic properties of both estimators are elucidated and the difference from the quantile estimators in time do- main is also indicated. We construct a testing procedure of quantile tests from the asymptotic distribution of the estimators and strong statistical power is shown in our numerical studies.
In this article, we consider an imputation method to handle missing response values based on semiparametric quantile regression estimation. In the proposed method, the missing response values are generated using the estimated conditional quantile regression function at given values of covariates. We adopt the generalized method of moments for estimation of parameters defined through a general estimation equation. We demonstrate that the proposed estimator, which combines both semiparametric quantile regression imputation and generalized method of moments, has competitive edge against some of t…
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 requi…
In this paper I develop a wild bootstrap procedure for cluster-robust inference in linear quantile regression models. I show that the bootstrap leads to asymptotically valid inference on the entire quantile regression process in a setting with a large number of small, heterogeneous clusters and provides consistent estimates of the asymptotic covariance function of that process. The proposed bootstrap procedure is easy to implement and performs well even when the number of clusters is much smaller than the sample size. An application to Project STAR data is provided.
This paper considers quantile model with grouped explanatory variables. In order to have the sparsity of the parameter groups but also the sparsity between two successive groups of variables, we propose and study an adaptive fused group LASSO quantile estimator. The number of variable groups can be fixed or divergent. We find the convergence rate under classical assumptions and we show that the proposed estimator satisfies the oracle properties.
The paper introduces an estimation method for flexible Bayesian quantile regression in ordinal (FBQROR) models i.e., an ordinal quantile regression where the error follows a generalized asymmetric Laplace (GAL) distribution. The GAL distribution, unlike the asymmetric Laplace (AL) distribution, allows to fix specific quantiles while simultaneously letting the mode, skewness and tails to vary. We also introduce the cumulative distribution function (necessary for constructing the likelihood) and the moment generating function of the GAL distribution. The algorithm is illustrated in multiple simu…
Quantile regression is often used when a comprehensive relationship between a response variable and one or more explanatory variables is desired. The traditional frequentists' approach to quantile regression has been well developed around asymptotic theories and efficient algorithms. However, not much work has been published under the Bayesian framework. One challenging problem for Bayesian quantile regression is that the full likelihood has no parametric forms. In this paper, we propose a Bayesian quantile regression method, the linearly interpolated density (LID) method, which uses a linear …
Genetic studies often involve quantitative traits. Identifying genetic features that influence quantitative traits can help to uncover the etiology of diseases. Quantile regression method considers the conditional quantiles of the response variable, and is able to characterize the underlying regression structure in a more comprehensive manner. On the other hand, genetic studies often involve high dimensional genomic features, and the underlying regression structure may be heterogeneous in terms of both effect sizes and sparsity. To account for the potential genetic heterogeneity, including the…
We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
A new multivariate concept of quantile, based on a directional version of Koenker and Bassett's traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow f…
In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest. The parameter function is estimated by combining ideas from f…
This paper deals with a linear model of regression on quantiles when the explanatory variable takes values in some functional space and the response is scalar. We propose a spline estimator of the functional coefficient that minimizes a penalized L1 type criterion. Then, we study the asymptotic behavior of this estimator. The penalization is of primary importance to get existence and convergence.
Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. (2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content irrespective of the underlying distribution; their graphs constitute nested center-outward quantile regression tubes. Empirical counterparts of these concepts are constructed, yielding interpretable empirical regions and contours wh…
A nonparametric procedure for robust regression estimation and for quantile regression is proposed which is completely data-driven and adapts locally to the regularity of the regression function. This is achieved by considering in each point M-estimators over different local neighbourhoods and by a local model selection procedure based on sequential testing. Non-asymptotic risk bounds are obtained, which yield rate-optimality for large sample asymptotics under weak conditions. Simulations for different univariate median regression models show good finite sample properties, also in comparison t…
In the present paper we investigate the predictive risk of possibly misspecified quantile regression functions. The in-sample risk is well-known to be an overly optimistic estimate of the predictive risk and we provide two relatively simple (asymptotic) characterizations of the associated bias, also called expected optimism. We propose estimates for the expected optimism and the predictive risk, and establish their uniform consistency under mild conditions. Our results hold for models of moderately growing size and allow the quantile function to be incorrectly specified. Empirical evidence fro…
The presence of \b{eta}-convergence in European regions is an important issue to be analyzed. In this paper, we adopt a quantile regression approach in analyzing economic convergence. While previous work has performed quantile regression at the national level, we focus on 187 European NUTS2 regions for the period 1981-2009 and use spatial quantile regression to account for spatial dependence.
In this paper, we establish a uniform error rate of a Bahadur representation for local polynomial estimators of quantile regression functions. The error rate is uniform over a range of quantiles, a range of evaluation points in the regressors, and over a wide class of probabilities for observed random variables. Most of the existing results on Bahadur representations for local polynomial quantile regression estimators apply to the fixed data generating process. In the context of testing monotonicity where the null hypothesis is of a complex composite hypothesis, it is particularly relevant to …
Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantile…
We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalized estimators such as the adaptive lasso yield sub-optimal rates if the coefficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specific problems of censored data and yield estimates with an optimal rate. In contrast to most of t…
It is well established that migratory birds in general have advanced their arrival times in spring, and in this paper we investigate potential ways of enhancing the level of detail in future phenological analyses. We perform single as well as multiple species analyses, using linear models on empirical quantiles, non-parametric quantile regression and likelihood-based parametric quantile regression with asymmetric Laplace distributed error terms. We conclude that non-parametric quantile regression appears most suited for single as well as multiple species analyses.
Quantile regression is a statistical method for estimating conditional quantiles of a response variable. In addition, for mean estimation, it is well known that quantile regression is more robust to outliers than $l_2$-based methods. By using the fused lasso penalty over a $K$-nearest neighbors graph, we propose an adaptive quantile estimator in a non-parametric setup. We show that the estimator attains optimal rate of $n^{-1/d}$ up to a logarithmic factor, under mild assumptions on the data generation mechanism of the $d$-dimensional data. We develop algorithms to compute the estimator and di…
Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model class. We derive an upper bound on the expected mean squared error of a ReLU network used to estimate any quantile conditional on a set of covariates. This upper bound only depends on the best possible approximation error, the number of layers in the network, and the number of nodes per layer. We further show upper bounds that are tight for two large classes of…
Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects. In this paper, we consider statistical inference for quantile regression with large-scale data in the "increasing dimension" regime. We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression. This method, which we refer to as {\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differ…
This paper studies the statistical properties of the group Lasso estimator for high dimensional sparse quantile regression models where the number of explanatory variables (or the number of groups of explanatory variables) is possibly much larger than the sample size while the number of variables in "active" groups is sufficiently small. We establish a non-asymptotic bound on the $\ell_{2}$-estimation error of the estimator. This bound explains situations under which the group Lasso estimator is potentially superior/inferior to the $\ell_{1}$-penalized quantile regression estimator in terms of…
In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap met…
Quantile regression is a tool for learning conditional distributions. In this paper we study quantile regression in the setting where a protected attribute is unavailable when fitting the model. This can lead to "unfair'' quantile estimators for which the effective quantiles are very different for the subpopulations defined by the protected attribute. We propose a procedure for adjusting the estimator on a heldout sample where the protected attribute is available. The main result of the paper is an empirical process analysis showing that the adjustment leads to a fair estimator for which the t…
We propose to smooth the entire objective function, rather than only the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also all…
In ordinary quantile regression, quantiles of different order are estimated one at a time. An alternative approach, which is referred to as quantile regression coefficients modeling (QRCM), is to model quantile regression coefficients as parametric functions of the order of the quantile. In this paper, we describe how the QRCM paradigm can be applied to longitudinal data. We introduce a two-level quantile function, in which two different quantile regression models are used to describe the (conditional) distribution of the within-subject response and that of the individual effects. We propose a…
Department of Mathematics University of California San Diego（加利福尼亚大学圣地亚哥分校数学系）
Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on new estimating equations that adapt to censoring and lead to quantile score whenever the data do not exhibit censoring. The proposed procedure named censored quantile regression f…
The widespread use of quantile regression methods depends crucially on the existence of fast algorithms. Despite numerous algorithmic improvements, the computation time is still non-negligible because researchers often estimate many quantile regressions and use the bootstrap for inference. We suggest two new fast algorithms for the estimation of a sequence of quantile regressions at many quantile indexes. The first algorithm applies the preprocessing idea of Portnoy and Koenker (1997) but exploits a previously estimated quantile regression to guess the sign of the residuals. This step allows f…
This paper investigates the identification of quantiles and quantile regression parameters when observations are set valued. We define the identification set of quantiles of random sets in a way that extends the definition of quantiles for regular random variables. We then give sharp characterization of this set by extending concepts from random set theory. For quantile regression parameters, we show that the identification set is characterized by a system of conditional moment inequalities. This characterization extends that of parametric quantile regression for regular random variables. Esti…
The Hong Kong Polytechnic University（香港理工大学）；The Chinese University of Hong Kong（香港中文大学）
We propose a nonparametric quantile regression method using deep neural networks with a rectified linear unit penalty function to avoid quantile crossing. This penalty function is computationally feasible for enforcing non-crossing constraints in multi-dimensional nonparametric quantile regression. We establish non-asymptotic upper bounds for the excess risk of the proposed nonparametric quantile regression function estimators. Our error bounds achieve optimal minimax rate of convergence for the Holder class, and the prefactors of the error bounds depend polynomially on the dimension of the pr…
We propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. We further extend our partial quantile covariance techniques to functional composite quantile regression (CQR) defining partial composite quantile covariance. There are three major contributions. (1) We define partial quantile covariance between two scalar variables through linear quantile regr…
The R package quantreg.nonpar implements nonparametric quantile regression methods to estimate and make inference on partially linear quantile models. quantreg.nonpar obtains point estimates of the conditional quantile function and its derivatives based on series approximations to the nonparametric part of the model. It also provides pointwise and uniform confidence intervals over a region of covariate values and/or quantile indices for the same functions using analytical and resampling methods. This paper serves as an introduction to the package and displays basic functionality of the functio…
As a competitive alternative to least squares regression, quantile regression is popular in analyzing heterogenous data. For quantile regression model specified for one single quantile level $τ$, major difficulties of semiparametric efficient estimation are the unavailability of a parametric efficient score and the conditional density estimation. In this paper, with the help of the least favorable submodel technique, we first derive the semiparametric efficient scores for linear quantile regression models that are assumed for a single quantile level, multiple quantile levels and all the quanti…
We consider a heteroscedastic regression model in which some of the regression coefficients are zero but it is not known which ones. Penalized quantile regression is a useful approach for analyzing such data. By allowing different covariates to be relevant for modeling conditional quantile functions at different quantile levels, it provides a more complete picture of the conditional distribution of a response variable than mean regression. Existing work on penalized quantile regression has been mostly focused on point estimation. Although bootstrap procedures have recently been shown to be eff…
We investigate different methods for regularizing quantile regression when predicting either a subset of quantiles or the full inverse CDF. We show that minimizing an expected pinball loss over a continuous distribution of quantiles is a good regularizer even when only predicting a specific quantile. For predicting multiple quantiles, we propose achieving the classic goal of non-crossing quantiles by using deep lattice networks that treat the quantile as a monotonic input feature, and we discuss why monotonicity on other features is an apt regularizer for quantile regression. We show that latt…
Quantile regression relates the quantile of the response to a linear predictor. For a discrete response distributions, like the Poission, Binomial and the negative Binomial, this approach is not feasible as the quantile function is not bijective. We argue to use a continuous model-aware interpolation of the quantile function, allowing for proper quantile inference while retaining model interpretation. This approach allows for proper uncertainty quantification and mitigates the issue of quantile crossing. Our reanalysis of hospitalisation data considered in Congdon (2017) shows the advantages o…
