Estimation and variable selection for generalized additive partial linear models

Wang, Li; Liu, Xiang; Liang, Hua; Carroll, Raymond J.

doi:10.1214/11-aos885

Cited by 122 publications

(86 citation statements)

References 34 publications

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“…Here we only list a few. See Härdle et al (2000), Härdle et al (2004), Ma and Yang (2011), Wang et al (2011), Lian (2012) and Guo et al (2013). However, the above mentioned authors only considered the problem of estimation and variable selection for independent data.…”

Section: Introductionmentioning

confidence: 99%

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

Yang

Guo

2015

Comput Stat

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This paper develops a robust and efficient estimation procedure for quantile partially linear additive models with longitudinal data, where the nonparametric components are approximated by B spline basis functions. The proposed approach can incorporate the correlation structure between repeated measures to improve estimation efficiency. Moreover, the new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations method for non-normal correlated random errors. However, the proposed estimating functions are non-smooth and non-convex. In order to reduce computational burdens, we apply the induced smoothing method for fast and accurate computation of the parameter estimates and its asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distribution of the estimators for the parametric components and the convergence rate of the estimators for the nonparametric functions. Furthermore, a variable selection procedure based on smooth-threshold estimating equations is developed to simultaneously identify non-zero parametric and nonparametric components. Finally, simulation studies have been conducted to evaluate the finite sample performance of the proposed method, and a real data example is analyzed to illustrate the application of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

Yang

Guo

2015

Comput Stat

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“…constructed by the sum of smooth functions of predictor variables, where it is common to use defined polynomials forn intervals known as "splines" [8][9][10]. For this reason, a function of this type loses its purely parametric nature and becomes a semi-parametric or non-parametric model [11]. In addition, GAMs can be applied without the compliance of independent regressors or a specific normal distribution shape of the sample [10].…”

Section: Study Areamentioning

confidence: 99%

Generalized Models: An Application to Identify Environmental Variables That Significantly Affect the Abundance of Three Tree Species

Antúnez

Hernández-Díaz

Wehenkel

et al. 2017

Forests

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Abstract:In defining the environmental preferences of plant species, statistical models are part of the essential tools in the field of modern ecology. However, conventional linear models require compliance with some parametric assumptions and if these requirements are not met, imply a serious limitation of the applied model. In this study, the effectiveness of linear and nonlinear generalized models was examined to identify the unitary effect of the principal environmental variables on the abundance of three tree species growing in the natural temperate forests of Oaxaca, Mexico. The covariates that showed a significant effect on the distribution of tree species were the maximum and minimum temperatures and the precipitation during specific periods. Results suggest that the generalized models, particularly smoothed models, were able to detect the increase or decrease of the abundance against changes in an environmental variable; they also revealed the inflection of the regression. In addition, these models allow partial characterization of the realized niche of a given species according to some specific variables, regardless of the type of relationship.

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“…By treating Z T β l as a covariate U l and letting X l ≡ 1 for all 1 ≤ l ≤ d, model (4) may be regarded as an additive model (Hastie and Tibshirani (1990) and Wang and Yang (2007)), and moreover by letting m l (·) ≡ m l for some l, it is a partially linear additive model (PLAM, Wang et al (2011) and ).…”

Section: Insert Figure 2 Herementioning

confidence: 99%

Varying Index Coefficient Models

Ma¹,

Song

2015

Journal of the American Statistical Association

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It has been a long history of utilizing interactions in regression analysis to investigate interactive effects of covariates on response variables. In this paper we aim to address two kinds of new challenges resulted from the inclusion of such highorder effects in the regression model for complex data. The first kind arises from a situation where interaction effects of individual covariates are weak but those of combined covariates are strong, and the other kind pertains to the presence of nonlinear interactive effects. Generalizing the single index coefficient regression model , we propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable. As a result, most of the existing semiparametric regression models are special cases of our proposed models. We develop a numerically stable and computationally fast estimation procedure utilizing both profile least squares method and local fitting. We establish both estimation consistency and asymptotic normality for the proposed estimators of index coefficients as well as the oracle property for the nonparametric function estimator. In addition, a generalized likelihood ratio test is provided to test for the existence of interaction effects or the existence of nonlinear interaction effects. Our models and estimation methods are illustrated by both simulation studies and an analysis of body fat dataset. Varying Index Coefficient Models 1Shujie Ma and Peter X.-K. Song AbstractIt has been a long history of utilizing interactions in regression analysis to investigate interactive effects of covariates on response variables. In this paper we aim to address two kinds of new challenges resulted from the inclusion of such high-order effects in the regression model for complex data. The first kind arises from a situation where interaction effects of individual covariates are weak but those of combined covariates are strong, and the other kind pertains to the presence of nonlinear interactive effects. Generalizing the single index coefficient regression model , we propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable. As a result, most of the existing semiparametric regression models are special cases of our proposed models. We develop a numerically stable and computationally fast estimation procedure utilizing both profile least squares method and local fitting. We establish both estimation consistency and asymptotic normality for the proposed estimators of index coefficients as well as the oracle property for the nonparametric function estimator. In addition, a generalized likelihood ratio test is provided to test for the existence of interaction effects or the existence of nonlinear interaction effects. Our models and estimation methods are illustrated by both simulation studies and an analysis of...

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Estimation and variable selection for generalized additive partial linear models

Cited by 122 publications

References 34 publications

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

Generalized Models: An Application to Identify Environmental Variables That Significantly Affect the Abundance of Three Tree Species

Varying Index Coefficient Models

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