Penalized quantile regression (QR) is widely used for studying the relationship between a response variable and a set of predictors under data heterogeneity in high-dimensional settings. Compared to penalized least squares, scalable algorithms for fitting penalized QR are lacking due to the non-differentiable piecewise linear loss function. To overcome the lack of smoothness, a recently proposed convolution-type smoothed method brings an interesting tradeoff between statistical accuracy and computational efficiency for both standard and penalized quantile regressions. In this paper, we propose a unified algorithm for fitting penalized convolution smoothed quantile regression with various commonly used convex penalties, accompanied by an R-language package conquer available from the Comprehensive R Archive Network. We perform extensive numerical studies to demonstrate the superior performance of the proposed algorithm over existing methods in both statistical and computational aspects. We further exemplify the proposed algorithm by fitting a fused lasso additive QR model on the world happiness data.
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…
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.
In this paper, we discuss a family of robust, high-dimensional regression models for quantile and composite quantile regression, both with and without an adaptive lasso penalty for variable selection. We reformulate these quantile regression problems and obtain estimators by applying the alternating direction method of multipliers (ADMM), majorize-minimization (MM), and coordinate descent (CD) algorithms. Our new approaches address the lack of publicly available methods for (composite) quantile regression, especially for high-dimensional data, both with and without regularization. Through simu…
We consider both $\ell _{0}$-penalized and $\ell _{0}$-constrained quantile regression estimators. For the $\ell _{0}$-penalized estimator, we derive an exponential inequality on the tail probability of excess quantile prediction risk and apply it to obtain non-asymptotic upper bounds on the mean-square parameter and regression function estimation errors. We also derive analogous results for the $\ell _{0}$-constrained estimator. The resulting rates of convergence are nearly minimax-optimal and the same as those for $\ell _{1}$-penalized and non-convex penalized estimators. Further, we charact…
We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates each regression coefficient and its corresponding subgradient simultaneously in each iteration. It combines the strengths of the coordinate descent and the semismooth Newton algorithm, and effectively solves the computational challenges posed by dimensionality and nonsmoothness. We establish the convergence properties of the algorithm. In addition, we p…
We consider a regression modeling of the quantiles of residual life, remaining lifetime at a specific time. We propose a smoothed induced version of the existing non-smooth estimating equations approaches for estimating regression parameters. The proposed estimating equations are smooth in regression parameters, so solutions can be readily obtained via standard numerical algorithms. Moreover, the smoothness in the proposed estimating equations enables one to obtain a robust sandwich-type covariance estimator of regression estimators aided by an efficient resampling method. To handle data subje…
In this paper, we consider binary response models with linear quantile restrictions. Considerably generalizing previous research on this topic, our analysis focuses on an infinite collection of quantile estimators. We derive a uniform linearisation for the properly standardized empirical quantile process and discover some surprising differences with the setting of continuously observed responses. Moreover, we show that considering quantile processes provides an effective way of estimating binary choice probabilities without restrictive assumptions on the form of the link function, heteroskedas…
Asgl: A Python Package for Penalized Linear and Quantile Regression
University of Columbia（哥伦比亚大学）；uc3m-Santander Big Data Institute（uc3m-桑坦德大数据研究所）；Universitat Politècnica de València（瓦伦西亚理工大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
This paper, investigates the conditional quantile estimation of a scalar random response and a functional random covariate (i.e. valued in some infinite-dimensional space) whenever {\it functional stationary ergodic data with random censorship} are considered. We introduce a kernel type estimator of the conditional quantile function. We establish the strong consistency with rate of this estimator as well as the asymptotic normality which induces a confidence interval that is usable in practice since it does not depend on any unknown quantity. An application to electricity peak demand interval …
This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on …
This article considers a linear model in a high dimensional data scenario. We propose a process which uses multiple loss functions both to select relevant predictors and to estimate parameters, and study its asymptotic properties. Variable selection is conducted by a procedure called "vote", which aggregates results from penalized loss functions. Using multiple objective functions separately simplifies algorithms and allows parallel computing, which is convenient and fast. As a special example we consider a quantile regression model, which optimally combines multiple quantile levels. We show t…
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…
$\ell_1$-penalized quantile regression is widely used for analyzing high-dimensional data with heterogeneity. It is now recognized that the $\ell_1$-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized $M$-estimation with strongly convex loss functions have been well studied, the extant literature on quantile regression is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non…
This paper investigates the effects of smoothed bootstrap iterations on coverage probabilities of smoothed bootstrap and bootstrap-t confidence intervals for population quantiles, and establishes the optimal kernel bandwidths at various stages of the smoothing procedures. The conventional smoothed bootstrap and bootstrap-t methods have been known to yield one-sided coverage errors of orders O(n^{-1/2}) and o(n^{-2/3}), respectively, for intervals based on the sample quantile of a random sample of size n. We sharpen the latter result to O(n^{-5/6}) with proper choices of bandwidths at the boots…
We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models. In these models the overall number of regressors $p$ is very large, possibly larger than the sample size $n$, but only $s$ of these regressors have non-zero impact on the conditional quantile of the response variable, where $s$ grows slower than $n$. We consider quantile regression penalized by the $\ell_1$-norm of coefficients ($\ell_1$-QR). First, we show that $\ell_1$-QR is consistent at the rate $\sqrt{s/n} \sqrt{\log p}$. The overall number of regres…
The standard quantile regression model assumes a linear relationship at the quantile of interest and that all variables are observed. We relax these assumptions by considering a partial linear model while allowing for missing linear covariates. To handle the potential bias caused by missing data we propose a weighted objective function using inverse probability weighting. Our work examines estimators using parametric and nonparametric estimates of the missing probability. For variable selection of the linear terms in the presence of missing data we consider a penalized and weighted objective f…
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…
The moment conditions or estimating equations for instrumental variables quantile regression involve the discontinuous indicator function. We instead use smoothed estimating equations (SEE), with bandwidth $h$. We show that the mean squared error (MSE) of the vector of the SEE is minimized for some $h>0$, leading to smaller asymptotic MSE of the estimating equations and associated parameter estimators. The same MSE-optimal $h$ also minimizes the higher-order type I error of a SEE-based $χ^2$ test and increases size-adjusted power in large samples. Computation of the SEE estimator also becomes …
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, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the …
In this article, we propose a penalized high dimensional semiparametric model average quantile prediction approach that is robust for forecasting the conditional quantile of the response. We consider a two-step estimation procedure. In the first step, we use a local linear regression approach to estimate the individual marginal quantile functions, and approximate the conditional quantile of the response by an affine combination of one-dimensional marginal quantile regression functions. In the second step, based on the nonparametric kernel estimates of the marginal quantile regression functions…
Thus far, likelihood-based interval estimates for quantiles have not been studied in the literature on interval censored case 2 data and partly interval censored data, and, in this context, the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval censored data. The fourth order expansion of the weighted empirica…
Censored quantile regression (CQR) has become a valuable tool to study the heterogeneous association between a possibly censored outcome and a set of covariates, yet computation and statistical inference for CQR have remained a challenge for large-scale data with many covariates. In this paper, we focus on a smoothed martingale-based sequential estimating equations approach, to which scalable gradient-based algorithms can be applied. Theoretically, we provide a unified analysis of the smoothed sequential estimator and its penalized counterpart in increasing dimensions. When the covariate dimen…
A central limit theorem (CLT) for the smoothed empirical spectral distribution of sample covariance matrices is established. Moreover, the CLTs for the smoothed quantiles of Marcenko and Pastur's law have been also developed.
This article deals with the analysis of high dimensional data that come from multiple sources (experiments) and thus have different possibly correlated responses, but share the same set of predictors. The measurements of the predictors may be different across experiments. We introduce a new regression approach with multiple quantiles to select those predictors that affect any of the responses at any quantile level and estimate the nonzero parameters. Our estimator is a minimizer of a penalized objective function, which aggregates the data from the different experiments. We establish model sele…
We study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signal…
In this paper, we consider a high-dimensional quantile regression model where the sparsity structure may differ between two sub-populations. We develop $\ell_1$-penalized estimators of both regression coefficients and the threshold parameter. Our penalized estimators not only select covariates but also discriminate between a model with homogeneous sparsity and a model with a change point. As a result, it is not necessary to know or pretest whether the change point is present, or where it occurs. Our estimator of the change point achieves an oracle property in the sense that its asymptotic dist…
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.
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…
Quantile regression has become a valuable tool to analyze heterogeneous covaraite-response associations that are often encountered in practice. The development of quantile regression methodology for high-dimensional covariates primarily focuses on examination of model sparsity at a single or multiple quantile levels, which are typically pre-specified ad hoc by the users. The resulting models may be sensitive to the specific choices of the quantile levels, leading to difficulties in interpretation and erosion of confidence in the results. In this article, we propose a new penalization framework…
Traditional Bayesian quantile regression relies on the Asymmetric Laplace distribution (ALD) mainly because of its satisfactory empirical and theoretical performances. However, the ALD displays medium tails and it is not suitable for data characterized by strong deviations from the Gaussian hypothesis. In this paper, we propose an extension of the ALD Bayesian quantile regression framework to account for fat-tails using the Skew Exponential Power (SEP) distribution. Beside having the $τ$-level quantile as parameter, the SEP distribution has an additional key parameter governing the decay of th…
This paper develops a two-part finite mixture quantile regression model for semi-continuous longitudinal data. The proposed methodology allows heterogeneity sources that influence the model for the binary response variable, to influence also the distribution of the positive outcomes. As is common in the quantile regression literature, estimation and inference on the model parameters are based on the Asymmetric Laplace distribution. Maximum likelihood estimates are obtained through the EM algorithm without parametric assumptions on the random effects distribution. In addition, a penalized versi…
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…
In this manuscript, we study quantile regression in partial functional linear model where response is scalar and predictors include both scalars and multiple functions. Wavelet basis are adopted to better approximate functional slopes while effectively detect local features. The sparse group lasso penalty is imposed to select important functional predictors while capture shared information among them. The estimation problem can be reformulated into a standard second-order cone program and then solved by an interior point method. We also give a novel algorithm by using alternating direction met…
This paper proposes a maximum-likelihood approach to jointly estimate marginal conditional quantiles of multivariate response variables in a linear regression framework. We consider a slight reparameterization of the Multivariate Asymmetric Laplace distribution proposed by Kotz et al (2001) and exploit its location-scale mixture representation to implement a new EM algorithm for estimating model parameters. The idea is to extend the link between the Asymmetric Laplace distribution and the well-known univariate quantile regression model to a multivariate context, i.e. when a multivariate depend…
The curse of dimensionality is a recognized challenge in nonparametric estimation. This paper develops a new L0-norm regularization approach to the convex quantile and expectile regressions for subset variable selection. We show how to use mixed integer programming to solve the proposed L0-norm regularization approach in practice and build a link to the commonly used L1-norm regularization approach. A Monte Carlo study is performed to compare the finite sample performances of the proposed L0-penalized convex quantile and expectile regression approaches with the L1-norm regularization approache…
This paper studies non-separable models with a continuous treatment when the dimension of the control variables is high and potentially larger than the effective sample size. We propose a three-step estimation procedure to estimate the average, quantile, and marginal treatment effects. In the first stage we estimate the conditional mean, distribution, and density objects by penalized local least squares, penalized local maximum likelihood estimation, and numerical differentiation, respectively, where control variables are selected via a localized method of L1-penalization at each value of the …
Quantile regression is a powerful statistical methodology that complements the classical linear regression by examining how covariates influence the location, scale, and shape of the entire response distribution and offering a global view of the statistical landscape. In this paper we propose a new quantile regression model for longitudinal data. The proposed approach incorporates the correlation structure between repeated measures to enhance the efficiency of the inference. In order to use the Newton-Raphson iteration method to obtain convergent estimates, the estimating functions are redefin…
