Doubly robust weighted composite quantile regression based on SCAD-L2

In this article, a robust variable selection procedure based on the weighted composite quantile regression (WCQR) is proposed. Compared with the composite quantile regression (CQR), WCQR is robust to heavy-tailed errors and outliers in the explanatory variables. For the choice of the weights in the WCQR, we employ a weighting scheme based on the principal component method. To select variables with grouping effect, we consider WCQR with SCAD-L₂ penalization. Furthermore, under some suitable assumptions, the theoretical properties, including the consistency and oracle property of the estimator, are established with a diverging number of parameters. In addition, we study the numerical performance of the proposed method in the case of ultrahigh-dimensional data. Simulation studies and real examples are provided to demonstrate the superiority of our method over the CQR method when there are outliers in the explanatory variables and/or the random error is from a heavy-tailed distribution.     

A note on the efficiency of composite quantile regression

Composite quantile regression (CQR) is motivated by the desire to have an estimator for linear regression models that avoids the breakdown of the least-squares estimator when the error variance is infinite, while having high relative efficiency even when the least-squares estimator is fully efficient. Here, we study two weighting schemes to further improve the efficiency of CQR, motivated by Jiang et al. [Oracle model selection for nonlinear models based on weighted composite quantile regression. Statist Sin. 2012;22:1479–1506]. In theory the two weighting schemes are asymptotically equivalent to each other and always result in more efficient estimators compared with CQR. Although the first weighting scheme is hard to implement, it sheds light on in what situations the improvement is expected to be large. A main contribution is to theoretically and empirically identify that standard CQR has good performance compared with weighted CQR only when the error density is logistic or close to logistic in shape, which was not noted in the literature.     

Focused information criterion for locally misspecified vector autoregressive models

This paper investigates the focused information criterion and plug-in average for vector autoregressive models with local-to-zero misspecification. These methods have the advantage of focusing on a quantity of interest rather than aiming at overall model fit. Any (su?ciently regular) function of the parameters can be used as a quantity of interest. We determine the asymptotic properties and elaborate on the role of the locally misspecified parameters. In particular, we show that the inability to consistently estimate locally misspecified parameters translates into suboptimal selection and averaging. We apply this framework to impulse response analysis. A Monte Carlo simulation study supports our claims.     

Weighted composite quantile regression method via empirical likelihood for non linear models

In this paper, we investigate empirical likelihood (EL) inferences via weighted composite quantile regression for non linear models. Under regularity conditions, we establish that the proposed empirical log-likelihood ratio is asymptotically chi-squared, and then the confidence intervals for the regression coefficients are constructed. The proposed method avoids estimating the unknown error density function involved in the asymptotic covariance matrix of the estimators. Simulations suggest that the proposed EL procedure is more efficient and robust, and a real data analysis is used to illustrate the performance.     

A weighted quantile regression for left-truncated and right-censored data

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. In this article, it is demonstrated that the estimating equation of Zhou [A weighted quantile regression for randomly truncated data, Comput. Stat. Data Anal. 55 (2011), pp. 554–566.] can be extended to analyse left-truncated and right-censored data. We evaluate the finite sample performance of the proposed estimators through simulation studies. The proposed estimator β?(q) is applied to the Veteran's Administration lung cancer data reported by Prentice [Exponential survival with censoring and explanatory variables, Biometrika 60 (1973), pp. 279–288].     

Focused and Model Average Estimation for Regression Analysis of Panel Count Data

Panel count data arise in many fields and a number of estimation procedures have been developed along with two procedures for variable selection. In this paper, we discuss model selection and parameter estimation together. For the former, a focused information criterion (FIC) is presented and for the latter, a frequentist model average (FMA) estimation procedure is developed. A main advantage, also the difference from the existing model selection methods, of the FIC is that it emphasizes the accuracy of the estimation of the parameters of interest, rather than all parameters. Further efficiency gain can be achieved by the FMA estimation procedure as unlike existing methods, it takes into account the variability in the stage of model selection. Asymptotic properties of the proposed estimators are established, and a simulation study conducted suggests that the proposed methods work well for practical situations. An illustrative example is also provided. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Empirical likelihood weighted composite quantile regression with partially missing covariates

This paper develops a novel weighted composite quantile regression (CQR) method for estimation of a linear model when some covariates are missing at random and the probability for missingness mechanism can be modelled parametrically. By incorporating the unbiased estimating equations of incomplete data into empirical likelihood (EL), we obtain the EL-based weights, and then re-adjust the inverse probability weighted CQR for estimating the vector of regression coefficients. Theoretical results show that the proposed method can achieve semiparametric efficiency if the selection probability function is correctly specified, therefore the EL weighted CQR is more efficient than the inverse probability weighted CQR. Besides, our algorithm is computationally simple and easy to implement. Simulation studies are conducted to examine the finite sample performance of the proposed procedures. Finally, we apply the new method to analyse the US news College data.     

Model averaging for M-estimation

M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results.     

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.     

Variable selection and weighted composite quantile estimation of regression parameters with left-truncated data

In this paper, we consider the weighted composite quantile regression for linear model with left-truncated data. The adaptive penalized procedure for variable selection is proposed. The asymptotic normality and oracle property of the resulting estimators are also established. Simulation studies are conducted to illustrate the finite sample performance of the proposed methods.     

General composite quantile regression: Theory and methods

Abstract

In this article, we propose a new regression method called general composite quantile regression (GCQR) which releases the unrealistic finite error variance assumption being imposed by the traditional least squares (LS) method. Unlike the recently proposed composite quantile regression (CQR) method, our proposed GCQR allows any continuous non-uniform density/weight function. As a result, determination of the number of uniform quantile positions is not required. Most importantly, the proposed GCQR criterion can be readily transformed to a linear programing problem, which substantially reduces the computing time. Our theoretical and empirical results show that the GCQR is generally efficient than the CQR and LS if the weight function is appropriately chosen. The oracle properties of the penalized GCQR are also provided. Our simulation results are consistent with the derived theoretical findings. A real data example is analyzed to demonstrate our methodologies.     

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.     

A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models

Just as frequentist hypothesis tests have been developed to check model assumptions, prior predictive p-values and other Bayesian p-values check prior distributions as well as other model assumptions. These model checks not only suffer from the usual threshold dependence of p-values, but also from the suppression of model uncertainty in subsequent inference. One solution is to transform Bayesian and frequentist p-values for model assessment into a fiducial distribution across the models. Averaging the Bayesian or frequentist posterior distributions with respect to the fiducial distribution can reproduce results from Bayesian model averaging or classical fiducial inference.     

Estimation and variable selection in single-index composite quantile regression

In this article, a new composite quantile regression estimation approach is proposed for estimating the parametric part of single-index model. We use local linear composite quantile regression (CQR) for estimating the nonparametric part of single-index model (SIM) when the error distribution is symmetrical. The weighted local linear CQR is proposed for estimating the nonparametric part of SIM when the error distribution is asymmetrical. Moreover, a new variable selection procedure is proposed for SIM. Under some regularity conditions, we establish the large sample properties of the proposed estimators. Simulation studies and a real data analysis are presented to illustrate the behavior of the proposed estimators.     

Forecasting time series of economic processes by model averaging across data frames of various lengths

This paper presents an extension of mean-squared forecast error (MSFE) model averaging for integrating linear regression models computed on data frames of various lengths. Proposed method is considered to be a preferable alternative to best model selection by various efficiency criteria such as Bayesian information criterion (BIC), Akaike information criterion (AIC), F-statistics and mean-squared error (MSE) as well as to Bayesian model averaging (BMA) and naïve simple forecast average. The method is developed to deal with possibly non-nested models having different number of observations and selects forecast weights by minimizing the unbiased estimator of MSFE. Proposed method also yields forecast confidence intervals with a given significance level what is not possible when applying other model averaging methods. In addition, out-of-sample simulation and empirical testing proves efficiency of such kind of averaging when forecasting economic processes.     

A new model selection procedure based on dynamic quantile regression

In this article, we propose a novel robust data-analytic procedure, dynamic quantile regression (DQR), for model selection. It is robust in the sense that it can simultaneously estimate the coefficients and the distribution of errors over a large collection of error distributions even those that are heavy-tailed and may not even possess variances or means; and DQR is easy to implement in the sense that it does not need to decide in advance which quantile(s) should be gathered. Asymptotic properties of related estimators are derived. Simulations and illustrative real examples are also given.     

Local composite quantile regression estimation of time-varying parameter vector for multidimensional time-inhomogeneous diffusion models

This paper is dedicated to the study of the composite quantile regression (CQR) estimations of time-varying parameter vectors for multidimensional diffusion models. Based on the local linear fitting for parameter vectors, we propose the local linear CQR estimations of the drift parameter vectors, and verify their asymptotic biases, asymptotic variances and asymptotic normality. Moreover, we discuss the asymptotic relative efficiency (ARE) of the local linear CQR estimations with respect to the local linear least-squares estimations. We obtain that the local estimations that we proposed are much more efficient than the local linear least-squares estimations. Simulation studies are constructed to show the performance of the estimations proposed.     

Model averaging based on rank

In this paper, we investigate model selection and model averaging based on rank regression. Under mild conditions, we propose a focused information criterion and a frequentist model averaging estimator for the focused parameters in rank regression model. Compared to the least squares method, the new method is not only highly efficient but also robust. The large sample properties of the proposed procedure are established. The finite sample properties are investigated via extensive Monte Claro simulation study. Finally, we use the Boston Housing Price Dataset to illustrate the use of the proposed rank methods.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.