On multivariate quantile regression analysis

This paper investigates the estimation of parameters in a multivariate quantile regression model when the investigator wants to evaluate the associated distribution function. It proposes a new directional quantile estimator with the following properties: (1) it applies to an arbitrary number of random variables; (2) it is equivalent to estimating the distribution function allowing for non-convex distribution contours; (3) it satisfies nice equivariance properties; (4) it has desirable statistical properties (i.e., consistency and asymptotic normality); and (5) its implementation involves a modest computational burden: our proposed estimator can be obtained by solving parametric linear programming problems. As such, this paper expands the range of applications of quantile estimation for multivariate regression models.     

Partially linear censored quantile regression

Censored regression quantile (CRQ) methods provide a powerful and flexible approach to the analysis of censored survival data when standard linear models are felt to be appropriate. In many cases however, greater flexibility is desired to go beyond the usual multiple regression paradigm. One area of common interest is that of partially linear models: one (or more) of the explanatory covariates are assumed to act on the response through a non-linear function. Here the CRQ approach of Portnoy (J Am Stat Assoc 98:1001–1012, 2003) is extended to this partially linear setting. Basic consistency results are presented. A simulation experiment and unemployment example justify the value of the partially linear approach over methods based on the Cox proportional hazards model and on methods not permitting nonlinearity.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

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.     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

Weighted composite quantile regression for partially linear varying coefficient models

Partially linear varying coefficient models (PLVCMs) with heteroscedasticity are considered in this article. Based on composite quantile regression, we develop a weighted composite quantile regression (WCQR) to estimate the non parametric varying coefficient functions and the parametric regression coefficients. The WCQR is augmented using a data-driven weighting scheme. Moreover, the asymptotic normality of proposed estimators for both the parametric and non parametric parts are studied explicitly. In addition, by comparing the asymptotic relative efficiency theoretically and numerically, WCQR method all outperforms the CQR method and some other estimate methods. To achieve sparsity with high-dimensional covariates, we develop a variable selection procedure to select significant parametric components for the PLVCM and prove the method possessing the oracle property. Both simulations and data analysis are conducted to illustrate the finite-sample performance of the proposed methods.     

Hierarchically penalized quantile regression

In many regression problems, predictors are naturally grouped. For example, when a set of dummy variables is used to represent categorical variables, or a set of basis functions of continuous variables is included in the predictor set, it is important to carry out a feature selection both at the group level and at individual variable levels within the group simultaneously. To incorporate the group and variables within-group information into a regularized model fitting, several regularization methods have been developed, including the Cox regression and the conditional mean regression. Complementary to earlier works, the simultaneous group and within-group variables selection method is examined in quantile regression. We propose a hierarchically penalized quantile regression, and show that the hierarchical penalty possesses the oracle property in quantile regression, as well as in the Cox regression. The proposed method is evaluated through simulation studies and a real data application.     

Structured kernel quantile regression

Quantile regression can provide more useful information on the conditional distribution of a response variable given covariates while classical regression provides informations on the conditional mean alone. In this paper, we propose a structured quantile estimation methodology in a nonparametric function estimation setup. Through the functional analysis of variance decomposition, the optimization of the proposed method can be solved using a series of quadratic and linear programmings. Our method automatically selects relevant covariates by adopting a lasso-type penalty. The performance of the proposed methodology is illustrated through numerical examples on both simulated and real data.     

Identification of Salmonella high risk pig-herds in Belgium by using semiparametric quantile regression

Summary.  Consumption of pork that is contaminated with Salmonella is an important source of human salmonellosis world wide. To control and prevent salmonellosis, Belgian pig-herds with high Salmonella infection burden are encouraged to take part in a control programme supporting the implementation of control measures. The Belgian government decided that only the 10% of pig-herds with the highest Salmonella infection burden (denoted high risk herds) can participate. To identify these herds, serological data reported as sample-to-positive ratios (SP-ratios) are collected. However, SP-ratios have an extremely skewed distribution and are heavily subject to confounding seasonal and animal age effects. Therefore, we propose to identify the 10% high risk herds by using semiparametric quantile regression with P -splines. In particular, quantile curves of animal SP-ratios are estimated as a function of sampling time and animal age. Then, pigs are classified into low and high risk animals with high risk animals having an SP-ratio that is larger than the corresponding estimated upper quantile. Finally, for each herd, the number of high risk animals is calculated as well as the beta–binomial p -value reflecting the hypothesis that the Salmonella infection burden is higher in that herd compared with the other herds. The 10% pig-herds with the lowest p -values are then identified as high risk herds. In addition, since high risk herds are supported to implement control measures, a risk factor analysis is conducted by using binomial generalized linear mixed models to investigate factors that are associated with decreased or increased Salmonella infection burden. Finally, since the choice of a specific upper quantile is to a certain extent arbitrary, a sensitivity analysis is conducted comparing different choices of upper quantiles.     

On eliminating inferior regression models

Consider a linear regression model with [p-1] predictor variables which is taken as the "true" model.The goal Is to select a subset of all possible reduced models such that all inferior models ‘to be defined’ are excluded with a guaranteed minimum probability.A procedure is proposed for which the exact evaluation of the probability of a correct decision 1s difficult; however, 1t is shown that the probability requirement can be met for sufficiently large sample size.Monte Carlo evaluation of the constant associated with the procedure and some ways to reduce the amount of computations Involved in the Implementation of the procedure are discussed.     

On quantile residuals in beta regression

Beta regression is often used to model the relationship between a dependent variable that assumes values on the open interval (0, 1) and a set of predictor variables. An important challenge in beta regression is to find residuals whose distribution is well approximated by the standard normal distribution. Two previous works compared residuals in beta regression, but the authors did not include the quantile residual. Using Monte Carlo simulation techniques, this article studies the behavior of certain residuals in beta regression in several scenarios. Overall, the results suggest that the distribution of the quantile residual is better approximated by the standard normal distribution than that of the other residuals in most scenarios. Three applications illustrate the effectiveness of the quantile residual.     

A weighted linear quantile regression

In this article, we introduce a new weighted quantile regression method. Traditionally, the estimation of the parameters involved in quantile regression is obtained by minimizing a loss function based on absolute distances with weights independent of explanatory variables. Specifically, we study a new estimation method using a weighted loss function with the weights associated with explanatory variables so that the performance of the resulting estimation can be improved. In full generality, we derive the asymptotic distribution of the weighted quantile regression estimators for any uniformly bounded positive weight function independent of the response. Two practical weighting schemes are proposed, each for a certain type of data. Monte Carlo simulations are carried out for comparing our proposed methods with the classical approaches. We also demonstrate the proposed methods using two real-life data sets from the literature. Both our simulation study and the results from these examples show that our proposed method outperforms the classical approaches when the relative efficiency is measured by the mean-squared errors of the estimators.     

On an adaptive transformation–retransformation estimate of multivariate location

An affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied. The work is primarily motivated by earlier work on different versions of the multivariate median and their properties. We explore an issue related to efficiency and equivariance that was originally raised by Bickel and subsequently investigated by Brown and Hettmansperger. Our estimate has better asymptotic performance than the vector of co-ordinatewise medians when the variables are substantially correlated. The finite sample performance of the estimate is investigated by using Monte Carlo simulations. Some examples are presented to demonstrate the effect of the adaptive transformation–retransformation strategy in the construction of multivariate location estimates for real data.     

Bayesian regression with multivariate linear splines

We present a Bayesian analysis of a piecewise linear model constructed by using basis functions which generalizes the univariate linear spline to higher dimensions. Prior distributions are adopted on both the number and the locations of the splines, which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty. Conditioning on the data produces a Bayes local linear model with distributions on both predictions and local linear parameters. The method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.     

Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

Variable selection in additive quantile regression using nonconcave penalty

This paper considers variable selection in additive quantile regression based on group smoothly clipped absolute deviation (gSCAD) penalty. Although shrinkage variable selection in additive models with least-squares loss has been well studied, quantile regression is sufficiently different from mean regression to deserve a separate treatment. It is shown that the gSCAD estimator can correctly identify the significant components and at the same time maintain the usual convergence rates in estimation. Simulation studies are used to illustrate our method.     

Adaptive elastic net-penalized quantile regression for variable selection

Abstract

There has been much attention on the high-dimensional linear regression models, which means the number of observations is much less than that of covariates. Considering the fact that the high dimensionality often induces the collinearity problem, in this article, we study the penalized quantile regression with the elastic net (EnetQR) that combines the strengths of the quadratic regularization and the lasso shrinkage. We investigate the weak oracle property of the EnetQR under mild conditions in the high dimensional setting. Moreover, we propose a two-step procedure, called adaptive elastic net quantile regression (AEnetQR), in which the weight vector in the second step is constructed from the EnetQR estimate in the first step. This two-step procedure is justified theoretically to possess the weak oracle property. The finite sample properties are performed through the Monte Carlo simulation and a real-data analysis.     

On partial linear additive isotonic regression

In this paper we discuss semiparametric additive isotonic regression models. We discuss the efficiency bound of the model and the least squares estimator under this model. We show that the ordinary least square estimator studied by Huang (2002) and Cheng (2009) for the semiparametric isotonic regression achieves the efficiency bound for the regular estimator when the true parameter belongs to the interior of the parameter space. We also show that the result by Cheng (2009) can be generalized to the case that the covariates are dependent on each other.