Estimation for the censored partially linear quantile regression models

In this article, we develop estimation procedures for partially linear quantile regression models, where some of the responses are censored by another random variable. The nonparametric function is estimated by basis function approximations. The estimation procedure is easy to implement through existing weighted quantile regression, and it requires no specification of the error distributions. We show the large-sample properties of the resulting estimates, the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the estimator of the functional component achieves the optimal convergence rate of the nonparametric function. The proposed method is studied via simulations and illustrated with the analysis of a primary biliary cirrhosis (BPC) data.     

Nonstationary nonlinear quantile regression

This study examines estimation and inference based on quantile regression for parametric nonlinear models with an integrated time series covariate. We first derive the limiting distribution of the nonlinear quantile regression estimator and then consider testing for parameter restrictions, when the regression function is specified as an asymptotically homogeneous function. We also study linear-in-parameter regression models when the regression function is given by integrable regression functions as well as asymptotically homogeneous regression functions. We, furthermore, propose a fully modified estimator to reduce the bias in the original estimator under a certain set of conditions. Finally, simulation studies show that the estimators behave well, especially when the regression error term has a fat-tailed distribution.     

Quantile regression in functional linear semiparametric model

This paper proposes nonparametric estimation methods for functional linear semiparametric quantile regression, where the conditional quantile of the scalar responses is modelled by both scalar and functional covariates and an additional unknown nonparametric function term. The slope function is estimated using the functional principal component basis and the nonparametric function is approximated by a piecewise polynomial function. The asymptotic distribution of the estimators of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. The asymptotic distribution of the estimator of the unknown nonparametric function is also established. Simulation studies are conducted to investigate the finite-sample performance of the proposed estimators. The proposed methodology is demonstrated by analysing a real data from ADHD-200 sample.     

Component selection in additive quantile regression models

Nonparametric additive models are powerful techniques for multivariate data analysis. Although many procedures have been developed for estimating additive components both in mean regression and quantile regression, the problem of selecting relevant components has not been addressed much especially in quantile regression. We present a doubly-penalized estimation procedure for component selection in additive quantile regression models that combines basis function approximation with a ridge-type penalty and a variant of the smoothly clipped absolute deviation penalty. We show that the proposed estimator identifies relevant and irrelevant components consistently and achieves the nonparametric optimal rate of convergence for the relevant components. We also provide an accurate and efficient computation algorithm to implement the estimator and demonstrate its performance through simulation studies. Finally, we illustrate our method via a real data example to identify important body measurements to predict percentage of body fat of an individual.     

Box–Cox power transformation unconditional quantile regressions with an application on wage inequality

This study proposes a semi-parametric estimation method, Box–Cox power transformation unconditional quantile regression, to estimate the impact of changes in the distribution of the explanatory variables on the unconditional quantile of the outcome variable. The proposed method consists of running a nonlinear regression of the recentered influence function (RIF) of the outcome variable on the explanatory variables. We also show the asymptotic properties of the proposed estimator and apply the estimation method to address an existing puzzle in labor economics–why the 50th/10th percentile wage gap has been falling in the USA since the late 1980s. Our results show that declining unionization can explain approximately 10% of the decline in the 50/10 wage gap in 1990–2000 and 23% in 2000–2010.     

Bootstrapping quantiles in a fixed design regression model with censored data

We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

ROBUST ESTIMATION OF SMALL‐AREA MEANS AND QUANTILES

Small‐area estimation techniques have typically relied on plug‐in estimation based on models containing random area effects. More recently, regression M‐quantiles have been suggested for this purpose, thus avoiding conventional Gaussian assumptions, as well as problems associated with the specification of random effects. However, the plug‐in M‐quantile estimator for the small‐area mean can be shown to be the expected value of this mean with respect to a generally biased estimator of the small‐area cumulative distribution function of the characteristic of interest. To correct this problem, we propose a general framework for robust small‐area estimation, based on representing a small‐area estimator as a functional of a predictor of this small‐area cumulative distribution function. Key advantages of this framework are that it naturally leads to integrated estimation of small‐area means and quantiles and is not restricted to M‐quantile models. We also discuss mean squared error estimation for the resulting estimators, and demonstrate the advantages of our approach through model‐based and design‐based simulations, with the latter using economic data collected in an Australian farm survey.     

Quantile regression based on a weighted approach under semi-competing risks data

In this article, we investigate the quantile regression analysis for semi-competing risks data in which a non-terminal event may be dependently censored by a terminal event. Due to the dependent censoring, the estimation of quantile regression coefficients on the non-terminal event becomes difficult. In order to handle this problem, we assume Archimedean Copula to specify the dependence of the non-terminal event and the terminal event. Portnoy [Censored regression quantiles. J Amer Statist Assoc. 2003;98:1001–1012] considered the quantile regression model under right-censoring data. We extend his approach to construct a weight function, and then impose the weight function to estimate the quantile regression parameter for the non-terminal event under semi-competing risks data. We also prove the consistency and asymptotic properties for the proposed estimator. According to the simulation studies, the performance of our proposed method is good. We also apply our suggested approach to analyse a real data.     

Adaptive composite quantile regressions and their asymptotic relative efficiency

The composite quantile regression (CQR for short) provides an efficient and robust estimation for regression coefficients. In this paper we introduce two adaptive CQR methods. We make two contributions to the quantile regression literature. The first is that, both adaptive estimators treat the quantile levels as realizations of a random variable. This is quite different from the classic CQR in which the quantile levels are typically equally spaced, or generally, are treated as fixed values. Because the asymptotic variances of the adaptive estimators depend upon the generic distribution of the quantile levels, it has the potential to enhance estimation efficiency of the classic CQR. We compare the asymptotic variance of the estimator obtained by the CQR with that obtained by quantile regressions at each single quantile level. The second contribution is that, in terms of relative efficiency, the two adaptive estimators can be asymptotically equivalent to the CQR method as long as we choose the generic distribution of the quantile levels properly. This observation is useful in that it allows to perform parallel distributed computing when the computational complexity issue arises for the CQR method. We compare the relative efficiency of the adaptive methods with respect to some existing approaches through comprehensive simulations and an application to a real-world problem.     

Linear censored quantile regression: A novel minimum-distance approach

In this article, we investigate a new procedure for the estimation of a linear quantile regression with possibly right-censored responses. Contrary to the main literature on the subject, we propose in this context to circumvent the formulation of conditional quantiles through the so-called “check” loss function that stems from the influential work of Koenker and Bassett (1978). Instead, our suggestion is here to estimate the quantile coefficients by minimizing an alternative measure of distance. In fact, our approach could be qualified as a generalization in a parametric regression framework of the technique consisting in inverting the conditional distribution of the response given the covariates. This is motivated by the knowledge that the main literature for censored data already relies on some nonparametric conditional distribution estimation as well. The ideas of effective dimension reduction are then exploited in order to accommodate for higher dimensional settings as well in this context. Extensive numerical results then suggest that such an approach provides a strongly competitive procedure to the classical approaches based on the check function, in fact both for complete and censored observations. From a theoretical prospect, both consistency and asymptotic normality of the proposed estimator for linear regression are obtained under classical regularity conditions. As a by-product, several asymptotic results on some “double-kernel” version of the conditional Kaplan–Meier distribution estimator based on effective dimension reduction, and its corresponding density estimator, are also obtained and may be of interest on their own. A brief application of our procedure to quasar data then serves to further highlight the relevance of the latter for quantile regression estimation with censored data.     

Quantile regression methods for left-truncated and right-censored data

Left-truncated and right-censored (LTRC) data are encountered frequently due to a prevalent cohort sampling in follow-up studies. Because of the skewness of the distribution of survival time, quantile regression is a useful alternative to the Cox's proportional hazards model and the accelerated failure time model for survival analysis. In this paper, we apply the quantile regression model to LTRC data and develops an unbiased estimating equation for regression coefficients. The proposed estimation methods use the inverse probabilities of truncation and censoring weighting technique. The resulting estimator is uniformly consistent and asymptotically normal. The finite-sample performance of the proposed estimation methods is also evaluated using extensive simulation studies. Finally, analysis of real data is presented to illustrate our proposed estimation methods.     

Consistent model identification of varying coefficient quantile regression with BIC tuning parameter selection

Quantile regression provides a flexible platform for evaluating covariate effects on different segments of the conditional distribution of response. As the effects of covariates may change with quantile level, contemporaneously examining a spectrum of quantiles is expected to have a better capacity to identify variables with either partial or full effects on the response distribution, as compared to focusing on a single quantile. Under this motivation, we study a general adaptively weighted LASSO penalization strategy in the quantile regression setting, where a continuum of quantile index is considered and coefficients are allowed to vary with quantile index. We establish the oracle properties of the resulting estimator of coefficient function. Furthermore, we formally investigate a Bayesian information criterion (BIC)-type uniform tuning parameter selector and show that it can ensure consistent model selection. Our numerical studies confirm the theoretical findings and illustrate an application of the new variable selection procedure.     

Fast Censored Linear Regression

Weighted log‐rank estimating function has become a standard estimation method for the censored linear regression model, or the accelerated failure time model. Well established statistically, the estimator defined as a consistent root has, however, rather poor computational properties because the estimating function is neither continuous nor, in general, monotone. We propose a computationally efficient estimator through an asymptotics‐guided Newton algorithm, in which censored quantile regression methods are tailored to yield an initial consistent estimate and a consistent derivative estimate of the limiting estimating function. We also develop fast interval estimation with a new proposal for sandwich variance estimation. The proposed estimator is asymptotically equivalent to the consistent root estimator and barely distinguishable in samples of practical size. However, computation time is typically reduced by two to three orders of magnitude for point estimation alone. Illustrations with clinical applications are provided.     

函数型变量倾斜分位回归模型及其应用

本文考虑函数型数据的结构特征,针对两类函数型变量分位回归模型（函数型因变量对标量自变量和函数型因变量对函数型自变量）,基于函数型倾斜分位曲线的定义构建新型函数型倾斜分位回归模型。对于第二类模型,本文分别考虑样条基函数对模型系数展开和函数型主成分基函数对函数型自变量展开,得到倾斜分位回归模型的基本形式。参数估计采用成分梯度Boosting算法最小化加权非对称损失函数,提高计算效率。在理论上证明了倾斜分位回归模型的系数估计量均服从渐近正态分布。模拟和实证研究结果显示,倾斜分位回归模型比已有的逐点分位回归模型具有更好的拟合效果。根据积分均方预测误差准则,本文提出的模型有一致较好的预测能力。     

Robust Estimation of Multivariate Linear Model Based on Depth Weighted Mean and Scatter

Based on the projection depth weighted mean and scatter estimation of the joint distribution of (x, y), we introduce a robust estimator of the regression coefficients for the multivariate linear model. The new estimator possesses desirable properties including affine invariance, Fisher consistency, and asymptotic normality. Also, we study the robustness of the estimator in terms of breakdown point and influence function. Extensive simulation studies are performed to investigate the finite sample behavior of robustness and efficiency. The methodology is illustrated with a real data example.     

Penalized weighted composite quantile regression in the linear regression model with heavy-tailed autocorrelated errors

In this paper, a penalized weighted composite quantile regression estimation procedure is proposed to estimate unknown regression parameters and autoregression coefficients in the linear regression model with heavy-tailed autoregressive errors. Under some conditions, we show that the proposed estimator possesses the oracle properties. In addition, we introduce an iterative algorithm to achieve the proposed optimization problem, and use a data-driven method to choose the tuning parameters. Simulation studies demonstrate that the proposed new estimation method is robust and works much better than the least squares based method when there are outliers in the dataset or the autoregressive error distribution follows heavy-tailed distributions. Moreover, the proposed estimator works comparably to the least squares based estimator when there are no outliers and the error is normal. Finally, we apply the proposed methodology to analyze the electricity demand dataset.     

Quantile estimation in the proportional hazards model of random censorship

The asymptotic theory is given for quantile estimation in the proportional hazards model of random censorship. In this model, the tail of the censoring distribution function is some power of the tail of the survival distribution function. The quantile estimator is based on the maximum likelihood estimator for the survival time distribution, due to Abdushukurov, Cheng and Lin.     

Statistical inference for the quintile share ratio

In recent years, the Quintile Share Ratio (or QSR) has become a very popular measure of inequality. In 2001, the European Council decided that income inequality in European Union member states should be described using two indicators: the Gini Index and the QSR. The QSR is generally defined as the ratio of the total income earned by the richest 20% of the population relative to that earned by the poorest 20%. Thus, it can be expressed using quantile shares, where a quantile share is the share of total income earned by all of the units up to a given quantile. The aim of this paper is to propose an improved methodology for the estimation and variance estimation of the QSR in a complex sampling design framework. Because the QSR is a non-linear function of interest, the estimation of its sampling variance requires advanced methodology. Moreover, a non-trivial obstacle in the estimation of quantile shares in finite populations is the non-unique definition of a quantile. Thus, two different conceptions of the quantile share are presented in the paper, leading us to two different estimators of the QSR. Regarding variance estimation, [Osier, 2006] and [Osier, 2009] proposed a variance estimator based on linearization techniques. However, his method involves Gaussian kernel smoothing of cumulative distribution functions. Our approach, also based on linearization, shows that no smoothing is needed. The construction of confidence intervals is discussed and a proposition is made to account for the skewness of the sampling distribution of the QSR. Finally, simulation studies are run to assess the relevance of our theoretical results.     

Change-point detection in a linear model by adaptive fused quantile method

A novel approach to quantile estimation in multivariate linear regression models with change-points is proposed: the change-point detection and the model estimation are both performed automatically, by adopting either the quantile-fused penalty or the adaptive version of the quantile-fused penalty. These two methods combine the idea of the check function used for the quantile estimation and the L₁ penalization principle known from the signal processing and, unlike some standard approaches, the presented methods go beyond typical assumptions usually required for the model errors, such as sub-Gaussian or normal distribution. They can effectively handle heavy-tailed random error distributions, and, in general, they offer a more complex view on the data as one can obtain any conditional quantile of the target distribution, not just the conditional mean. The consistency of detection is proved and proper convergence rates for the parameter estimates are derived. The empirical performance is investigated via an extensive comparative simulation study and practical utilization is demonstrated using a real data example.