Testing for covariate balance using quantile regression and resampling methods

Consistency of propensity score matching estimators hinges on the propensity score's ability to balance the distributions of covariates in the pools of treated and non-treated units. Conventional balance tests merely check for differences in covariates’ means, but cannot account for differences in higher moments. For this reason, this paper proposes balance tests which test for differences in the entire distributions of continuous covariates based on quantile regression (to derive Kolmogorov–Smirnov and Cramer–von-Mises–Smirnov-type test statistics) and resampling methods (for inference). Simulations suggest that these methods are very powerful and capture imbalances related to higher moments when conventional balance tests fail to do so.     

Weighted composite quantile regression analysis for nonignorable missing data using nonresponse instrument

Efficient statistical inference on nonignorable missing data is a challenging problem. This paper proposes a new estimation procedure based on composite quantile regression (CQR) for linear regression models with nonignorable missing data, that is applicable even with high-dimensional covariates. A parametric model is assumed for modelling response probability, which is estimated by the empirical likelihood approach. Local identifiability of the proposed strategy is guaranteed on the basis of an instrumental variable approach. A set of data-based adaptive weights constructed via an empirical likelihood method is used to weight CQR functions. The proposed method is resistant to heavy-tailed errors or outliers in the response. An adaptive penalisation method for variable selection is proposed to achieve sparsity with high-dimensional covariates. Limiting distributions of the proposed estimators are derived. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies. An application to the ACTG 175 data is analysed.     

Distribution estimation with auxiliary information for missing data

There is much literature on statistical inference for distribution under missing data, but surprisingly very little previous attention has been paid to missing data in the context of estimating distribution with auxiliary information. In this article, the auxiliary information with missing data is proposed. We use Zhou, Wan and Wang's method (2008) to mitigate the effects of missing data through a reformulation of the estimating equations, imputed through a semi-parametric procedure. Whence we can estimate distribution and the τth quantile of the distribution by taking auxiliary information into account. Asymptotic properties of the distribution estimator and corresponding sample quantile are derived and analyzed. The distribution estimators based on our method are found to significantly outperform the corresponding estimators without auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

Quantile regression for competing risks analysis under case-cohort design

The case-cohort design brings cost reduction in large cohort studies. In this paper, we consider a nonlinear quantile regression model for censored competing risks under the case-cohort design. Two different estimation equations are constructed with or without the covariates information of other risks included, respectively. The large sample properties of the estimators are obtained. The asymptotic covariances are estimated by using a fast resampling method, which is useful to consider further inferences. The finite sample performance of the proposed estimators is assessed by simulation studies. Also a real example is used to demonstrate the application of the proposed methods.     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

Estimation and variable selection for partially functional linear models

In this paper, a new estimation procedure based on composite quantile regression and functional principal component analysis (PCA) method is proposed for the partially functional linear regression models (PFLRMs). The proposed estimation method can simultaneously estimate both the parametric regression coefficients and functional coefficient components without specification of the error distributions. The proposed estimation method is shown to be more efficient empirically for non-normal random error, especially for Cauchy error, and almost as efficient for normal random errors. Furthermore, based on the proposed estimation procedure, we use the penalized composite quantile regression method to study variable selection for parametric part in the PFLRMs. Under certain regularity conditions, consistency, asymptotic normality, and Oracle property of the resulting estimators are derived. Simulation studies and a real data analysis are conducted to assess the finite sample performance of the proposed methods.     

Estimation and inference of combining quantile and least-square regressions with missing data

In this paper, we consider how to incorporate quantile information to improve estimator efficiency for regression model with missing covariates. We combine the quantile information with least-squares normal equations and construct an unbiased estimating equations (EEs). The lack of smoothness of the objective EEs is overcome by replacing them with smooth approximations. The maximum smoothed empirical likelihood (MSEL) estimators are established based on inverse probability weighted (IPW) smoothed EEs and their asymptotic properties are studied under some regular conditions. Moreover, we develop two novel testing procedures for the underlying model. The finite-sample performance of the proposed methodology is examined by simulation studies. A real example is used to illustrate our methods.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

Invited Discussion Paper Resampling Methods in Sample Surveys

This article reviews the applications of three resampling methods, the jackknife, the balanced repeated replication, and the bootstrap, in sample surveys. The sampling design under consideration is a stratified multistage sampling design. We discuss the implementation of the resampling methods; for example, the construction of balanced repeated replications and approximated balanced repeated replication estimators; four modified bootstrap algorithms to generate bootstrap samples; and three different ways of applying the resampling methods in the presence of imputed missing values. Asymptotic properties of the resampling estimators are discussed for two types of important survey estimators, functions of weighted averages and sample quantiles.     

Weighted quantile regression and testing for varying-coefficient models with randomly truncated data

This paper develops a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data. In order to handle the truncated data, the random weights are introduced and the weighted quantile regression (WQR) estimators for nonparametric functions are proposed. To achieve nice efficiency properties, we further develop a weighted composite quantile regression (WCQR) estimation method for nonparametric functions in varying-coefficient models. The asymptotic properties both for the proposed WQR and WCQR estimators are established. In addition, we propose a novel bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms. The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.     

Weighted quantile regression with missing covariates using empirical likelihood

This paper proposes an empirical likelihood-based weighted (ELW) quantile regression approach for estimating the conditional quantiles when some covariates are missing at random. The proposed ELW estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness is correctly specified. The limiting covariance matrix of the ELW estimator can be estimated by a resampling technique, which does not involve nonparametric density estimation or numerical derivatives. Simulation results show that the ELW method works remarkably well in finite samples. A real data example is used to illustrate the proposed ELW method.     

Smoothed empirical likelihood for quantile regression models with response data missing at random

This paper studies smoothed quantile linear regression models with response data missing at random. Three smoothed quantile empirical likelihood ratios are proposed first and shown to be asymptotically Chi-squared. Then, the confidence intervals for the regression coefficients are constructed without the estimation of the asymptotic covariance. Furthermore, a class of estimators for the regression parameter is presented to derive its asymptotic distribution. Simulation studies are conducted to assess the finite sample performance. Finally, a real-world data set is analyzed to illustrated the effectiveness of the proposed methods.     

The generalized odd log-logistic family of distributions: properties,regression models and applications

We propose a new class of continuous distributions with two extra shape parameters named the generalized odd log-logistic family of distributions. The proposed family contains as special cases the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, two entropy measures and order statistics are obtained. We derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behaviour of the estimators by means of Monte Carlo simulations. We introduce the log-odd log-logistic Weibull regression model with censored data based on the odd log-logistic-Weibull distribution. The importance of the new family is illustrated using three real data sets. These applications indicate that this family can provide better fits than other well-known classes of distributions. The beauty and importance of the proposed family lies in its ability to model different types of real data.     

Efficient Robust Estimation for Linear Models with Missing Response at Random

Coefficient estimation in linear regression models with missing data is routinely carried out in the mean regression framework. However, the mean regression theory breaks down if the error variance is infinite. In addition, correct specification of the likelihood function for existing imputation approach is often challenging in practice, especially for skewed data. In this paper, we develop a novel composite quantile regression and a weighted quantile average estimation procedure for parameter estimation in linear regression models when some responses are missing at random. Instead of imputing the missing response by randomly drawing from its conditional distribution, we propose to impute both missing and observed responses by their estimated conditional quantiles given the observed data and to use the parametrically estimated propensity scores to weigh check functions that define a regression parameter. Both estimation procedures are resistant to heavy‐tailed errors or outliers in the response and can achieve nice robustness and efficiency. Moreover, we propose adaptive penalization methods to simultaneously select significant variables and estimate unknown parameters. Asymptotic properties of the proposed estimators are carefully investigated. An efficient algorithm is developed for fast implementation of the proposed methodologies. We also discuss a model selection criterion, which is based on an IC_Q ‐type statistic, to select the penalty parameters. The performance of the proposed methods is illustrated via simulated and real data sets.     

Moderate deviations for quantile regression processes

This paper mainly discusses the asymptotic properties of quantile regression processes. In view of the exponential tightness and convexity argument, we prove the quantile regression estimators satisfy the functional moderate deviation principle. This method can be extended to a fair range of different statistical estimation problems such as quantile regression estimators with bridge penalized functions.     

Randomized quantile regression estimation for heteroskedastic non parametric model

In this paper, we propose robust randomized quantile regression estimators for the mean and (condition) variance functions of the popular heteroskedastic non parametric regression model. Unlike classical approaches which consider quantile as a fixed quantity, our method treats quantile as a uniformly distributed random variable. Our proposed method can be employed to estimate the error distribution, which could significantly improve prediction results. An automatic bandwidth selection scheme will be discussed. Asymptotic properties and relative efficiencies of the proposed estimators are investigated. Our empirical results show that the proposed estimators work well even for random errors with infinite variances. Various numerical simulations and two real data examples are used to demonstrate our methodologies.     

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.     

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.     

B-spline estimation for partially linear varying coefficient composite quantile regression models

Abstract

In this article, a new composite quantile regression estimation (CQR) approach is proposed for partially linear varying coefficient models (PLVCM) under composite quantile loss function with B-spline approximations. The major advantage of the proposed procedures over the existing ones is easy to implement using existing software, and it requires no specification of the error distributions. Under the regularity conditions, the consistency and asymptotic normality of the estimators are also derived. Finally, a simulation study and a real data application are undertaken to assess the finite sample performance of the proposed estimation procedure.     

Multiple-output quantile regression through optimal quantization

A new nonparametric quantile regression method based on the concept of optimal quantization was developed recently and was showed to provide estimators that often dominate their classical, kernel-type, competitors. In the present work, we extend this method to multiple-output regression problems. We show how quantization allows approximating population multiple-output regression quantiles based on halfspace depth. We prove that this approximation becomes arbitrarily accurate as the size of the quantization grid goes to infinity. We also derive a weak consistency result for a sample version of the proposed regression quantiles. Through simulations, we compare the performances of our estimators with (local constant and local bilinear) kernel competitors. The results reveal that the proposed quantization-based estimators, which are local constant in nature, outperform their kernel counterparts and even often dominate their local bilinear kernel competitors. The various approaches are also compared on artificial and real data.