Semiparametric Partial Linear Models Research Articles

Semiparametric partial linear modeling of risk factors for ear infections: the Early Childhood Longitudinal Study

The Early Childhood Longitudinal Study–Kindergarten Class of 2010–2011 (ECLS-K:2011) ascertained timing of ear infections within age specified intervals and parent's/caregiver's report of medically diagnosed hearing loss. In this nationally representative, school-based sample of children followed from kindergarten entry through fifth grade, academic performance in reading, mathematics, and science was assessed longitudinally. Prior investigations of this ECLS-K:2011 cohort showed that age has a non-linear, monotonically increasing functional relationship with academic performance. Because of this knowledge, a semiparametric partial linear model is proposed, in which the effect of age is modeled by an unknown monotonically increasing function along with other regression parameters. The parameters are estimated by a semiparametric maximum likelihood estimator. A test of a constant effect of age is also proposed. Simulation studies are conducted to evaluate the performance of the proposed method, as compared with the commonly used linear model; the former outperforms the latter based on several criteria. We then analyzed ECLS-K:2011 data to compare results of the partial linear parametric model estimation with that of classical linear regression models.

Identification for partially linear regression model with autoregressive errors

The semiparametric partial linear models are often used in real data analysis for its flexibility and parsimony. Statistical inference of this model is restricted with two conditions: (i) the linear and nonlinear parts are known in advance, (ii) the errors are independent. However, in practice, this is unreasonable to artificially determine which subset of variables have linear effect on the response and which have nonlinear effect. In addition, the assumption of errors being independent may be incorrect for time series data. Therefore, it is of great interest to develop some efficient methods to distinguish linear components from nonlinear ones with correlated errors. In this paper, we develop a method for identifying linear and nonlinear components, and estimate the coefficients of error structure. The performance of the proposed method is examined by simulation study and analyses a real data set for an illustration.

Estimation in partial linear model with spline modal function

Semiparametric partial linear models are often used in real data analysis for their flexibility and parsimony. A well-known example of semiparametric partial linear models is the partially linear varying coefficient model. Statistical inference of this model is restricted to a condition that the components of varying and constant coefficients are known in advance. However, in practice, this is not known which subset of variables have constant or varying effect on the response. Therefore, it is of great interest to develop some efficient methods to distinguish constant components from varying ones. In this article, we propose a robust method for simultaneously structure identification and variable selection in varying coefficient models based on modal regression, which is robust with respect to non-normal errors and outliers in the response. The proposed procedure can achieve robustness and efficiency by using a bandwidth parameter. The performance of the proposed method is examined by a simulation study and real data analysis to show its capabilities when the error distribution is varied.

A semiparametric multivariate partially linear model: A difference approach

A multivariate semiparametric partial linear model for both fixed and random design cases is considered. In either case, the model is analyzed using a difference sequence approach. The linear component is estimated based on the differences of observations and the functional component is estimated using a multivariate Nadaraya–Watson kernel smoother of the residuals of the linear fit. We show that both components can be asymptotically estimated as well as if the other component were known. The estimator of the linear component is shown to be asymptotically normal and efficient in the fixed design case if the length of the difference sequence used goes to infinity at a certain rate. The functional component estimator is shown to be rate optimal if the Lipschitz smoothness index exceeds half the dimensionality of the functional component argument. We also develop a test for linear combinations of regression coefficients whose asymptotic power does not depend on the functional component. All of the proposed procedures are easy to implement. Finally, numerical performance of all the procedures is studied using simulated data.

Minimax rate of convergence for an estimator of the functional component in a semiparametric multivariate partially linear model

A multivariate semiparametric partial linear model for both fixed and random design cases is considered. Earlier, in Brown et al. (2014), the model has been analyzed using a difference sequence approach. In particular, the functional component has been estimated using a multivariate Nadaraya–Watson kernel smoother of the residuals of the linear fit. Moreover, this functional component estimator has been shown to be rate optimal if the Lipschitz smoothness index exceeds half the dimensionality of the functional component domain. In the current manuscript, we take this research further and show that, for both fixed and random designs, the rate achieved is the minimax rate under both risk at a point and the L 2 risk. The result is achieved by proving lower bounds on both pointwise risk and the L 2 risk of possible estimators of the functional component.

Semiparametric partial Linear quantile regression of longitudinal data with time varying coefficients and informative observation times

Regression analysis of longitudinal data has been a popular topic in many fields for long time. However, only limited research exists for the case where observation times may be informative and for quantile regression of longitudinal data. In particular, to our knowledge, there does not exist any established method for quantile regression of longitudinal data with informative observation times, the focus of this paper. More specifically, we will discuss this problem and present a semiparametric partial linear model with time-varying coefficients. For estimation, B-splines are used to approximate the time-varying coefficients and in addition to the estimation approach, model checking and selection procedures are also provided. The latter can be used to determine the covariates that indeed have time-varying effects on the longitudinal process of interest. The proposed method could identify the underlying true model structure and estimate the parameters simultaneously. Also we establish the convergence rate of the proposed estimators and the asymptotic normality of the estimated time-independent regression parameters. For the assessment of the finite sample performance of the proposed methods, an extensive simulation study is conducted and suggests that they work well for practical situations. Finally they are applied to a set of longitudinal medical cost data on chronic heart failure patients that motivated this study.

A simple estimator for partial linear regression with endogenous nonparametric variables

We propose a simple kernel estimator for semiparametric partial linear models with endogeneity in the nonparametric function. Compared to the existing backfitting estimator, our estimator is notationally simpler and relatively easier to implement. We also discuss data-driven bandwidth selection to implement this estimator in practice. Monte Carlo exercises show that the finite sample performance of these two estimators is similar.

Asymptotic efficiency of ridge estimator in linear and semiparametric linear models

The linear model with a growing number of predictors arises in many contemporary scientific endeavor. In this article, we consider the commonly used ridge estimator in linear models. We propose analyzing the ridge estimator for a finite sample size n and a growing dimension p . The existence and asymptotic normality of the ridge estimator are established under some regularity conditions when p → ∞ . It also occurs that a strictly linear model is inadequate when some of the relations are believed to be of certain linear form while others are not easily parameterized, and thus a semiparametric partial linear model is considered. For these semiparametric partial linear models with p > n , we develop a procedure to estimate the linear coefficients as if the nonparametric part is not present. The asymptotic efficiency of the proposed estimator for the linear component is studied for p → ∞ . It is shown that the proposed estimator of the linear component asymptotically performs very well.

A difference based approach to the semiparametric partial linear model

A commonly used semiparametric partial linear model is con- sidered. We propose analyzing this model using a difference based approach. The procedure estimates the linear component based on the differences of the observations and then estimates the nonparametric component by ei- ther a kernel or a wavelet thresholding method using the residuals of the linear fit. It is shown that both the estimator of the linear component and the estimator of the nonparametric component asymptotically perform as well as if the other component were known. The estimator of the linear com- ponent is asymptotically efficient and the estimator of the nonparametric component is asymptotically rate optimal. A test for linear combinations of the regression coefficients of the linear component is also developed. Both the estimation and the testing procedures are easily implementable. Nu- merical performance of the procedure is studied using both simulated and real data. In particular, we demonstrate our method in an analysis of an attitude data set as well as a data set from the Framingham Heart Study.

Partial Linear Models for Longitudinal Data Based on Quadratic Inference Functions

Abstract.In this paper, we consider improved estimating equations for semiparametric partial linear models (PLM) for longitudinal data, or clustered data in general. We approximate the non‐parametric function in the PLM by a regression spline, and utilize quadratic inference functions (QIF) in the estimating equations to achieve a more efficient estimation of the parametric part in the model, even when the correlation structure is misspecified. Moreover, we construct a test which is an analogue to the likelihood ratio inference function for inferring the parametric component in the model. The proposed methods perform well in simulation studies and real data analysis conducted in this paper.

Sieve likelihood ratio inference on general parameter space

In this paper, a theory on sieve likelihood ratio inference on general parameter spaces (including infinite dimensional) is studied. Under fairly general regularity conditions, the sieve log-likelihood ratio statistic is proved to be asymptotically χ 2 distributed, which can be viewed as a generalization of the well-known Wilks' theorem. As an example, a semiparametric partial linear model is investigated.