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Data are heterogeneous when recorded in different time regimes or taken from multiple sources to some degree. Varying-coefficient models or mixture models are suitable for solving this type of problem. On one hand, existing models are quite complicated and computationally cumbersome especially for large-scale data. On the other hand, common effects among different data sources are unknown. Additionally, some existing models are unable to search the non linear relationship between response and covariates. To address these challenges, we aim at estimating common effects about non parametric regression when data are heterogeneous. Our proposed estimation method is based on basis function expansion. Adaptive basis series and fixed basis series are considered, respectively. We exploit maximin aggregation technique to get a simple non linear model, also the common effects, from all possible grouped data. The mean squared error and asymptotic distribution of the estimator are investigated. Simulation studies and real-data analysis are conducted to verify the efficiency of the estimation procedure. Comparing with ordinary least square estimator and averaging ordinary least square estimator, our proposed estimator can reduce the complexity of data sources and is more robust. [ABSTRACT FROM AUTHOR]

This article considers the quantile regression approach for partially linear spatial autoregressive models with possibly varying coefficients. B-spline is employed for the approximation of varying coefficients. The instrumental variable quantile regression approach is employed for parameter estimation. The rank score tests are developed for hypotheses on the coefficients, including the hypotheses on the non-varying coefficients and the constancy of the varying coefficients. The asymptotic properties of the proposed estimators and test statistics are both established. Monte Carlo simulations are conducted to study the finite sample performance of the proposed method. Analysis of a real data example is presented for illustration. [ABSTRACT FROM AUTHOR]

Modern spatial temporal data are often collected from sensor networks. Missing data problems are common to this kind of data. Making accurate imputation is important for many applications. In the unsupervised setting, one technique is to minimize the rank of a tensor or matrix. If we add related covariates, can we get more accurate imputation results? To address this, we transform the original sensor×time measurements to high order tensors by adding additional temporal dimensions and then integrate tensor regression with tensor completion using nuclear norm penalty. One advantage is we can simultaneously estimate parameters and impute missing values due to clear spatial consistency for near-sited spatial-temporal data. The proposed method doesn't assume missing mechanism of the response. Theoretical properties of the proposed estimator are investigated. Simulation studies and real data analysis are conducted to verify the efficiency of the estimation procedure. [ABSTRACT FROM AUTHOR]

Time dynamic varying coefficient models play an important role in applications of biology, medicine, environment, finance, etc. Traditional methods, such as kernel smoothing and spline smoothing, are popular. But explicit expressions are unavailable using these methods, and the convergence rate of coefficient function estimators is slow. To address these problems, we expand the varying component with appropriate basis functions. And then we solve a sparse regression problem via a sequential thresholded least-squares estimator. The "parameterization" leads to explicit expressions and fast computation speed. Convergence of the sequential thresholded least squares algorithm is guaranteed. The asymptotic distribution of the coefficient function estimator is derived under certain assumptions. Our simulation shows the proposed method has higher precision and computing speed. Finally, our proposed method is applied to the study of PM 2.5 concentration in Beijing. We analyze the relationship between PM 2.5 and other impact factors. [ABSTRACT FROM AUTHOR]

In longitudinal data studies, a conjunct feature of the multitudinous statistical methods is to depict the conditional mean of outcome variable. However, the conditional mean is not the best measure of centrality which may be affected by outliers. As an available alterative, composite quantile regression (CQR) can result in robust estimation results even for non-normal error distributions in regression analysis instead of the least square estimation (LSE). In this paper, we develop a weighted CQR of longitudinal mixed model from a likelihood framework based on the composite asymmetric Laplace distribution (CALD). Using the mixture representation of the CALD, we establish the joint hierarchical likelihood of the model and achieve the iterative weighted least square estimators of unknown parameters via the MCEM (Monte Carlo Expectation Maximization) algorithm. Finally, the developed procedures are illustrated by Monte Carlo simulations and an AIDS (Acquired Immune Deficiency Syndrome) data analysis. [ABSTRACT FROM AUTHOR]

This paper investigates the quantile regression estimation for spatial error models with possibly varying coefficients. The local polynomial fitting scheme is employed to approximate the varying coefficients. The rank-based score test is developed for hypotheses on the model and the constancy of the varying coefficients. The asymptotic properties of the proposed estimators and test statistics are both established. Monte Carlo simulations are conducted to study the finite sample performance of the proposed method. Analysis of a real data example is presented for illustration. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In longitudinal data analysis, many data features are frequently encountered in practice. For example, variates are usually measured with censoring, substantial errors and non-normal feature, etc. In such case, it is impossible to give a precise result by conducting statistical inference of separate analysis. The joint modeling may be a considerable alternative. The superiority of joint modeling is that it can implement a simultaneous analysis for longitudinal mixed models with random effects, censoring, nor-normal errors, errors in covariates and heavy-tailed feature by pooling all data information together. However, most of traditional modeling methods aim at depicting the average variation of outcome variable conditionally on covariates, which may result in non-robust estimation results when suffering outliers or non-normal errors. Quantile regression provides an attractive alternative to model longitudinal data with multiple data features, which can draw a complete picture of the conditional distributions of outcome variable and bring out robust estimation results. In this paper, we conduct the likelihood-based joint quantile regression for longitudinal mixed models by accounting for the above multiple data features simultaneously. Based on the asymmetric Laplace distribution (ALD), the Monte Carlo Expectation-Maximization (MCEM) algorithm is employed to address the estimation problem. Finally, some Monte Carlo simulations are conducted and an AIDS data analysis is provided to illustrate the developed procedures. [ABSTRACT FROM AUTHOR]

Time lag effect exists widely in the course of economic operation. Some economic variables are affected not only by various factors in the current period but also by various factors in the past and even their own past values. As a class of dynamical models, autoregressive distributed lag (ARDL) models are frequently used to conduct dynamic regression analysis. In this paper, we are interested in the quantile regression (QR) modeling of the ARDL model in a dynamic framework. By combining the working likelihood of asymmetric Laplace distribution (ALD) with the expectation–maximization (EM) algorithm into the considered ARDL model, the iterative weighted least square estimators (IWLSE) are derived. Some Monte Carlo simulations are implemented to evaluate the performance of the proposed estimation method. A dataset of the consumption of electricity by residential customers is analyzed to illustrate the application. [ABSTRACT FROM AUTHOR]

This paper considers the quantile regression model with both individual fixed effect and time period effect for general spatial panel data. Fixed effects quantile regression estimators based on instrumental variable method will be proposed. Asymptotic properties of the proposed estimators will be developed. Simulations are conducted to study the performance of the proposed method. We will illustrate our methodologies using a cigarettes demand data set. [ABSTRACT FROM AUTHOR]

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as 'oracle'. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption. [ABSTRACT FROM AUTHOR]

Quantile regression (QR) is a natural alternative for depicting the impact of covariates on the conditional distributions of a outcome variable instead of the mean. In this paper, we investigate Bayesian regularized QR for the linear models with autoregressive errors. LASSO-penalized type priors are forced on regression coefficients and autoregressive parameters of the model. Gibbs sampler algorithm is employed to draw the full posterior distributions of unknown parameters. Finally, the proposed procedures are illustrated by some simulation studies and applied to a real data analysis of the electricity consumption. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients and the variance components of panel data regression models with complete panels. The PB pivot variables are proposed based on sufficient statistics of the parameters. On the other hand, we also derive generalized inferences and improved generalized inferences for variance components in this article. Some simulation results are presented to compare the performance of the PB approaches with the generalized inferences. Our studies show that the PB approaches perform satisfactorily for various sample sizes and parameter configurations, and the performance of PB approaches is mostly the same as that of generalized inferences with respect to the expected lengths and powers. The PB inferences have almost exact coverage probabilities and Type I error rates. Furthermore, the PB procedure can be simply carried out by a few simulation steps, and the derivation is easier to understand and to be extended to the incomplete panels. Finally, the proposed approaches are illustrated by using a real data example. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

In this article, we consider a generalized linear partially varying-coefficient model for longitudinal data analysis. A local quasi-likelihood method is proposed to estimate the constant-coefficient and varying-coefficient functions simultaneously based on the local polynomial kernel regression. The corresponding standard error estimates are derived. Large sample properties are investigated. The proposed methodologies are demonstrated by extensive simulation studies and a real example. [ABSTRACT FROM PUBLISHER]

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies. [ABSTRACT FROM PUBLISHER]

The varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example. [ABSTRACT FROM AUTHOR]

In biological, medical, and social sciences, multilevel structures are very common. Hierarchical models that take the dependencies among subjects within the same level are necessary. In this article, we introduce a semiparametric hierarchical composite quantile regression model for hierarchical data. This model (i) keeps the easy interpretability of the simple parametric model; (ii) retains some of the flexibility of the complex non parametric model; (iii) relaxes the assumptions that the noise variances and higher-order moments exist and are finite; and (iv) takes the dependencies among subjects within the same hierarchy into consideration. We establish the asymptotic properties of the proposed estimators. Our simulation results show that the proposed method is more efficient than the least-squares-based method for many non normally distributed errors. We illustrate our methodology with a real biometric data set. [ABSTRACT FROM PUBLISHER]

In this article, a transmuted linear exponential distribution is developed that generalizes the linear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al. Some statistical properties of the proposed distribution such as moments, quantiles, and the failure rate function are investigated. The maximum likelihood estimators of unknown parameters are also discussed and a real data analysis is carried out to illustrate the superiority of the proposed distribution. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

The Type-II progressive censoring scheme has become very popular for analyzing lifetime data in reliability and survival analysis. However, no published papers address parameter estimation under progressive Type-II censoring for the mixed exponential distribution (MED), which is an important model for reliability and survival analysis. This is the problem that we address in this paper. It is noted that maximum likelihood estimation of unknown parameters cannot be obtained in closed form due to the complicated log-likelihood function. We solve this problem by using the EM algorithm. Finally, we obtain closed form estimates of the model. The proposed methods are illustrated by both some simulations and a case analysis. [ABSTRACT FROM PUBLISHER]

This article aims to put forward a new method to solve the linear quantile regression problems based on EM algorithm using a location-scale mixture of the asymmetric Laplace error distribution. A closed form of the estimator of the unknown parameter vector β based on EM algorithm, is obtained. In addition, some simulations are conducted to illustrate the performance of the proposed method. Simulation results demonstrate that the proposed algorithm performs well. Finally, the classical Engel data is fitted and the Bootstrap confidence intervals for estimators are provided. [ABSTRACT FROM PUBLISHER]

In industrial life tests, reliability analysis and clinical trials, the type-II progressive censoring methodology, which allows for random removals of the remaining survival units at each failure time, has become quite popular for analyzing lifetime data. Parameter estimation under progressively type-II censored samples for many common lifetime distributions has been investigated extensively. However, how to estimate unknown parameters of the mixed distribution models under progressive type-II censoring schemes is still a challenging and interesting problem. Based on progressively type-II censored samples, this paper addresses the estimation problem of mixed generalized exponential distributions. In addition, it is observed that the maximum-likelihood estimates (MLEs) cannot be easily obtained in closed form due to the complexity of the likelihood function. Thus, we make good use of the expectation-maximization algorithm to obtain the MLEs. Finally, some simulations are implemented in order to show the performance of the proposed method under finite samples and a case analysis is illustrated. [ABSTRACT FROM PUBLISHER]

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained. [ABSTRACT FROM PUBLISHER]

Let {X1, ..., Xn} and {Y1, ..., Ym} be two samples of independent and identically distributed observations with common continuous cumulative distribution functions F(x)=P(X≤x) and G(y)=P(Y≤y), respectively. In this article, we would like to test the no quantile treatment effect hypothesis H0: F=G. We develop a bootstrap quantile-treatment-effect test procedure for testing H0 under the location-scale shift model. Our test procedure avoids the calculation of the check function (which is non-differentiable at the origin and makes solving the quantile effects difficult in typical quantile regression analysis). The limiting null distribution of the test procedure is derived and the procedure is shown to be consistent against a broad family of alternatives. Simulation studies show that our proposed test procedure attains its type I error rate close to the pre-chosen significance level even for small sample sizes. Our test procedure is illustrated with two real data sets on the lifetimes of guinea pigs from a treatment-control experiment. [ABSTRACT FROM AUTHOR]

AbstractObtaining excellent prediction accuracy in the high-dimensional partially linear model is particularly important. However, it is difficult to achieve due to the complex relationship between nonparametric covariates and the response. Irrelevant covariates and unimportant data also commonly attenuate the prediction performance of the model. Further, high-dimensional data analysis is challenging for modern statistical studies. To overcome these difficulties, we propose the Double Sparsity Garrotized Kernel Machine (DSGKM) method with an efficient algorithm and its adjusted version for prediction. Specifically, we estimate the nonparametric components using the kernel machine technique, and impose L1-norm penalties simultaneously to select relevant covariates and retain the representative data in the final model. Besides, the convergence analysis of the adjusted algorithm is conducted. The advantages of our method are: (i) to sufficiently capture the complex relationship between nonparametric covariates and the response; (ii) to identify relevant covariates and select representative data; and (iii) to achieve higher computational efficiency, especially the situations when both parametric and nonparametric components are high-dimensional. Results on both simulated and real data show that the proposed method outperforms existing methods, even when outliers exist. [ABSTRACT FROM AUTHOR]