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.     

Nonlinear semiparametric AR(1) model with skew-symmetric innovations

In this paper, we expand a first-order nonlinear autoregressive (AR) model with skew normal innovations. A semiparametric method is proposed to estimate a nonlinear part of model by using the conditional least squares method for parametric estimation and the nonparametric kernel approach for the AR adjustment estimation. Then computational techniques for parameter estimation are carried out by the maximum likelihood (ML) approach using Expectation-Maximization (EM) type optimization and the explicit iterative form for the ML estimators are obtained. Furthermore, in a simulation study and a real application, the accuracy of the proposed methods is verified.     

A new estimation procedure for a partially nonlinear model via a mixed‐effects approach

The authors consider the estimation of the parametric component of a partially nonlinear semiparametric regression model whose nonparametric component is viewed as a nuisance parameter. They show how estimation can proceed through a nonlinear mixed‐effects model approach. They prove that under certain regularity conditions, the proposed estimate is consistent and asymptotically Gaussian. They investigate its finite‐sample properties through simulations and illustrate its use with data on the relation between the photosynthetically active radiation and the net ecosystem‐atmosphere exchange of carbon dioxide.     

Copula‐based semiparametric analysis for time series data with detection limits

The analysis of time series data with detection limits is challenging due to the high‐dimensional integral involved in the likelihood. Existing methods are either computationally demanding or rely on restrictive parametric distributional assumptions. We propose a semiparametric approach, where the temporal dependence is captured by parametric copula, while the marginal distribution is estimated non‐parametrically. Utilizing the properties of copulas, we develop a new copula‐based sequential sampling algorithm, which provides a convenient way to calculate the censored likelihood. Even without full parametric distributional assumptions, the proposed method still allows us to efficiently compute the conditional quantiles of the censored response at a future time point, and thus construct both point and interval predictions. We establish the asymptotic properties of the proposed pseudo maximum likelihood estimator, and demonstrate through simulation and the analysis of a water quality data that the proposed method is more flexible and leads to more accurate predictions than Gaussian‐based methods for non‐normal data. The Canadian Journal of Statistics 47: 438–454; 2019 © 2019 Statistical Society of Canada     

Nonlinear semiparametric autoregressive model with finite mixtures of scale mixtures of skew normal innovations

We propose data generating structures which can be represented as the nonlinear autoregressive models with single and finite mixtures of scale mixtures of skew normal innovations. This class of models covers symmetric/asymmetric and light/heavy-tailed distributions, so provide a useful generalization of the symmetrical nonlinear autoregressive models. As semiparametric and nonparametric curve estimation are the approaches for exploring the structure of a nonlinear time series data set, in this article the semiparametric estimator for estimating the nonlinear function of the model is investigated based on the conditional least square method and nonparametric kernel approach. Also, an Expectation–Maximization-type algorithm to perform the maximum likelihood (ML) inference of unknown parameters of the model is proposed. Furthermore, some strong and weak consistency of the semiparametric estimator in this class of models are presented. Finally, to illustrate the usefulness of the proposed model, some simulation studies and an application to real data set are considered.     

Analysis of generalized semiparametric mixed varying‐coefficients models for longitudinal data

The generalized semiparametric mixed varying‐coefficient effects model for longitudinal data can accommodate a variety of link functions and flexibly model different types of covariate effects, including time‐constant, time‐varying and covariate‐varying effects. The time‐varying effects are unspecified functions of time and the covariate‐varying effects are nonparametric functions of a possibly time‐dependent exposure variable. A semiparametric estimation procedure is developed that uses local linear smoothing and profile weighted least squares, which requires smoothing in the two different and yet connected domains of time and the time‐dependent exposure variable. The asymptotic properties of the estimators of both nonparametric and parametric effects are investigated. In addition, hypothesis testing procedures are developed to examine the covariate effects. The finite‐sample properties of the proposed estimators and testing procedures are examined through simulations, indicating satisfactory performances. The proposed methods are applied to analyze the AIDS Clinical Trial Group 244 clinical trial to investigate the effects of antiretroviral treatment switching in HIV‐infected patients before and after developing the T215Y antiretroviral drug resistance mutation. The Canadian Journal of Statistics 47: 352–373; 2019 © 2019 Statistical Society of Canada     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.     

Efficient Estimation of the Partly Linear Additive Hazards Model with Current Status Data

This paper focuses on efficient estimation, optimal rates of convergence and effective algorithms in the partly linear additive hazards regression model with current status data. We use polynomial splines to estimate both cumulative baseline hazard function with monotonicity constraint and nonparametric regression functions with no such constraint. We propose a simultaneous sieve maximum likelihood estimation for regression parameters and nuisance parameters and show that the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound. In addition, we show that rates of convergence for the estimators of nonparametric functions are optimal. We implement the proposed estimation through a backfitting algorithm on generalized linear models. We conduct simulation studies to examine the finite‐sample performance of the proposed estimation method and present an analysis of renal function recovery data for illustration.     

Estimation of semi-varying coefficient models with nonstationary regressors

We study a semivarying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.     

Evaluating functional covariate‐environment interactions in the Cox regression model

Children exposed to mixtures of endocrine disrupting compounds such as phthalates are at high risk of experiencing significant friction in their growth and sexual maturation. This article is primarily motivated by a study that aims to assess the toxicants‐modified effects of risk factors related to the hazards of early or delayed onset of puberty among children living in Mexico City. To address the hypothesis of potential nonlinear modification of covariate effects, we propose a new Cox regression model with multiple functional covariate‐environment interactions, which allows covariate effects to be altered nonlinearly by mixtures of exposed toxicants. This new class of models is rather flexible and includes many existing semiparametric Cox models as special cases. To achieve efficient estimation, we develop the global partial likelihood method of inference, in which we establish key large‐sample results, including estimation consistency, asymptotic normality, semiparametric efficiency and the generalized likelihood ratio test for both parameters and nonparametric functions. The proposed methodology is examined via simulation studies and applied to the analysis of the motivating data, where maternal exposures to phthalates during the third trimester of pregnancy are found to be important risk modifiers for the age of attaining the first stage of puberty. The Canadian Journal of Statistics 47: 204–221; 2019 © 2019 Statistical Society of Canada     

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.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Semiparametric Estimation of Risk–Return Relationships

This article proposes semiparametric generalized least-squares estimation of parametric restrictions between the conditional mean and the conditional variance of excess returns given a set of parametric factors. A distinctive feature of our estimator is that it does not require a fully parametric model for the conditional mean and variance. We establish consistency and asymptotic normality of the estimates. The theory is nonstandard due to the presence of estimated factors. We provide sufficient conditions for the estimated factors not to have an impact in the asymptotic standard error of estimators. A simulation study investigates the finite sample performance of the estimates. Finally, an application to the CRSP value-weighted excess returns highlights the merits of our approach. In contrast to most previous studies using nonparametric estimates, we find a positive and significant price of risk in our semiparametric setting.     

On inference for a semiparametric partially linear regression model with serially correlated errors

The authors consider a semiparametric partially linear regression model with serially correlated errors. They propose a new way of estimating the error structure which has the advantage that it does not involve any nonparametric estimation. This allows them to develop an inference procedure consisting of a bandwidth selection method, an efficient semiparametric generalized least squares estimator of the parametric component, a goodness‐of‐fit test based on the bootstrap, and a technique for selecting significant covariates in the parametric component. They assess their approach through simulation studies and illustrate it with a concrete application.     

On Semiparametric Mode Regression Estimation

It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean, and median. In this article, the relation between mode, mean, and median regression functions is assumed to follow a simple parametric model. We propose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and median forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regression functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model.     

Bayesian inference for semiparametric regression using a Fourier representation

This paper presents the Bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components. The nonparametric component is represented with a Fourier series where the Fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates. The rate of decay controls the smoothness of the response function. The posterior analysis automatically selects the amount of smoothing that is coherent with the model and data. Posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non-specific alternative. The Bayes estimator's mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression.     

Semiparametric regression under long-range dependent errors

In this paper, we consider a semiparametric regression model under long-range dependent errors. By approximating the nonparametric component by a finite series sum, we construct consistent estimators for both parametric and nonparametric components. Meanwhile, convergence rates for the consistent estimators are also investigated. Additionally, an optimal truncation parameter selection procedure is proposed.     

Liu-type estimator in semiparametric partially linear additive models

Partially linear additive model is useful in statistical modelling as a multivariate nonparametric fitting technique. This paper considers statistical inference for the semiparametric model in the presence of multicollinearity. Based on the profile least-squares (PL) approach and Liu estimation method, we propose a PL Liu estimator for the parametric component. When some additional linear restrictions on the parametric component are available, the corresponding restricted Liu estimator for the parametric component is constructed. The properties of the proposed estimators are derived. Some simulations are conducted to assess the performance of the proposed procedures and the results are satisfactory. Finally, a real data example is analysed.     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

Dimension reduced kernel estimation for distribution function with incomplete data

This work focuses on the estimation of distribution functions with incomplete data, where the variable of interest Y has ignorable missingness but the covariate X is always observed. When X is high dimensional, parametric approaches to incorporate X—information is encumbered by the risk of model misspecification and nonparametric approaches by the curse of dimensionality. We propose a semiparametric approach, which is developed under a nonparametric kernel regression framework, but with a parametric working index to condense the high dimensional X—information for reduced dimension. This kernel dimension reduction estimator has double robustness to model misspecification and is most efficient if the working index adequately conveys the X—information about the distribution of Y. Numerical studies indicate better performance of the semiparametric estimator over its parametric and nonparametric counterparts. We apply the kernel dimension reduction estimation to an HIV study for the effect of antiretroviral therapy on HIV virologic suppression.