Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.     

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.     

Robust rank-based variable selection in double generalized linear models with diverging number of parameters under adaptive Lasso

We propose a robust rank-based estimation and variable selection in double generalized linear models when the number of parameters diverges with the sample size. The consistency of the variable selection procedure and asymptotic properties of the resulting estimators are established under appropriate selection of tuning parameters. Simulations are performed to assess the finite sample performance of the proposed estimation and variable selection procedure. In the presence of gross outliers, the proposed method is showing that the variable selection method works better. For practical application, a real data application is provided using nutritional epidemiology data, in which we explore the relationship between plasma beta-carotene levels and personal characteristics (e.g. age, gender, fat, etc.) as well as dietary factors (e.g. smoking status, intake of cholesterol, etc.).     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

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.     

NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors. In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.     

Empirical Likelihood for Nonparametric Components in Additive Partially Linear Models

Empirical likelihood-based inference for the nonparametric components in additive partially linear models is investigated. An empirical likelihood approach to construct the confidence intervals of the nonparametric components is proposed when the linear covariate is measured with and without errors. We show that the proposed empirical log-likelihood ratio is asymptotically standard chi-squared without requiring the undersmoothing of the nonparametric components. Then, it can be directly used to construct the confidence intervals for the nonparametric functions. A simulation study indicates that, compared with a normal approximation-based approach, the proposed method works better in terms of coverage probabilities and widths of the pointwise confidence intervals.     

Nonparametric Additive Instrumental Variable Estimator: A Group Shrinkage Estimation Perspective

In this article, we study a nonparametric approach regarding a general nonlinear reduced form equation to achieve a better approximation of the optimal instrument. Accordingly, we propose the nonparametric additive instrumental variable estimator (NAIVE) with the adaptive group Lasso. We theoretically demonstrate that the proposed estimator is root-n consistent and asymptotically normal. The adaptive group Lasso helps us select the valid instruments while the dimensionality of potential instrumental variables is allowed to be greater than the sample size. In practice, the degree and knots of B-spline series are selected by minimizing the BIC or EBIC criteria for each nonparametric additive component in the reduced form equation. In Monte Carlo simulations, we show that the NAIVE has the same performance as the linear instrumental variable (IV) estimator for the truly linear reduced form equation. On the other hand, the NAIVE performs much better in terms of bias and mean squared errors compared to other alternative estimators under the high-dimensional nonlinear reduced form equation. We further illustrate our method in an empirical study of international trade and growth. Our findings provide a stronger evidence that international trade has a significant positive effect on economic growth.     

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.     

Two-step variable selection in partially linear additive models with time series data

Lots of semi-parametric and nonparametric models are used to fit nonlinear time series data. They include partially linear time series models, nonparametric additive models, and semi-parametric single index models. In this article, we focus on fitting time series data by partially linear additive model. Combining the orthogonal series approximation and the adaptive sparse group LASSO regularization, we select the important variables between and within the groups simultaneously. Specially, we propose a two-step algorithm to obtain the grouped sparse estimators. Numerical studies show that the proposed method outperforms LASSO method in both fitting and forecasting. An empirical analysis is used to illustrate the methodology.     

Additive model selection

We study sparse high dimensional additive model fitting via penalization with sparsity-smoothness penalties. We review several existing algorithms that have been developed for this problem in the recent literature, highlighting the connections between them, and present some computationally efficient algorithms for fitting such models. Furthermore, using reasonable assumptions and exploiting recent results on group LASSO-like procedures, we take advantage of several oracle results which yield asymptotic optimality of estimators for high-dimensional but sparse additive models. Finally, variable selection procedures are compared with some high-dimensional testing procedures available in the literature for testing the presence of additive components.     

Sensitivity analysis of partially linear models with response missing at random

This article investigates case-deletion influence analysis via Cook’s distance and local influence analysis via conformal normal curvature for partially linear models with response missing at random. Local influence approach is developed to assess the sensitivity of parameter and nonparametric estimators to various perturbations such as case-weight, response variable, explanatory variable, and parameter perturbations on the basis of semiparametric estimating equations, which are constructed using the inverse probability weighted approach, rather than likelihood function. Residual and generalized leverage are also defined. Simulation studies and a dataset taken from the AIDS Clinical Trials are used to illustrate the proposed methods.     

Conditional VAR and Expected Shortfall: A New Functional Approach

We estimate two well-known risk measures, the value-at-risk (VAR) and the expected shortfall, conditionally to a functional variable (i.e., a random variable valued in some semi(pseudo)-metric space). We use nonparametric kernel estimation for constructing estimators of these quantities, under general dependence conditions. Theoretical properties are stated whereas practical aspects are illustrated on simulated data: nonlinear functional and GARCH(1,1) models. Some ideas on bandwidth selection using bootstrap are introduced. Finally, an empirical example is given through data of the S&P 500 time series.     

Penalized empirical likelihood inference for sparse additive hazards regression with a diverging number of covariates

High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by applications in high-throughput genomic data analysis. In this paper, we propose a class of regularization methods, integrating both the penalized empirical likelihood and pseudoscore approaches, for variable selection and estimation in sparse and high-dimensional additive hazards regression models. When the number of covariates grows with the sample size, we establish asymptotic properties of the resulting estimator and the oracle property of the proposed method. It is shown that the proposed estimator is more efficient than that obtained from the non-concave penalized likelihood approach in the literature. Based on a penalized empirical likelihood ratio statistic, we further develop a nonparametric likelihood approach for testing the linear hypothesis of regression coefficients and constructing confidence regions consequently. Simulation studies are carried out to evaluate the performance of the proposed methodology and also two real data sets are analyzed.     

Additive Nonparametric Regression in the Presence of Endogenous Regressors

In this article we consider nonparametric estimation of a structural equation model under full additivity constraint. We propose estimators for both the conditional mean and gradient which are consistent, asymptotically normal, oracle efficient, and free from the curse of dimensionality. Monte Carlo simulations support the asymptotic developments. We employ a partially linear extension of our model to study the relationship between child care and cognitive outcomes. Some of our (average) results are consistent with the literature (e.g., negative returns to child care when mothers have higher levels of education). However, as our estimators allow for heterogeneity both across and within groups, we are able to contradict many findings in the literature (e.g., we do not find any significant differences in returns between boys and girls or for formal versus informal child care). Supplementary materials for this article are available online.     

Semiparametric Estimation with Profile Algorithm for Longitudinal Binary Data

This article considers analyzing longitudinal binary data semiparametrically and proposing GEE-Smoothing spline in the estimation of parametric and nonparametric components. The method is an extension of the parametric generalized estimating equation to semiparametric. The nonparametric component is estimated by smoothing spline approach, i.e., natural cubic spline. We use profile algorithm in the estimation of both parametric and nonparametric components. Properties of the estimators are evaluated by simulation.     

Estimation and variable selection for generalised partially linear single-index models

In this paper, we study the problem of estimation and variable selection for generalised partially linear single-index models based on quasi-likelihood, extending existing studies on variable selection for partially linear single-index models to binary and count responses. To take into account the unit norm constraint of the index parameter, we use the ‘delete-one-component’ approach. The asymptotic normality of the estimates is demonstrated. Furthermore, the smoothly clipped absolute deviation penalty is added for variable selection of parameters both in the nonparametric part and the parametric part, and the oracle property of the variable selection procedure is shown. Finally, some simulation studies are carried out to illustrate the finite sample performance.     

Variable selection for partially varying coefficient single-index model

In this paper, we consider the problem of variable selection for partially varying coefficient single-index model, and present a regularized variable selection procedure by combining basis function approximations with smoothly clipped absolute deviation penalty. The proposed procedure simultaneously selects significant variables in the single-index parametric components and the nonparametric coefficient function components. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. Finite sample performance of the proposed method is illustrated by a simulation study and real data analysis.     

Nonparametric regression for dependent data in the errors-in-variables problem

We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.