Estimation for a partial-linear single-index model

Wang, Jane Ling; Xue, Liugen; Zhu, Lixing; Chong, Yun Sam

doi:10.1214/09-aos712

Cited by 171 publications

(86 citation statements)

References 39 publications

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“…There is also a great interest in developing partial-linear single-index models via the integration of single-index models with linear regression models (Carroll et al, 1997; Wang et al, 2010). Most earlier references focus on univariate response observed from cross-sectional studies (Xia et al, 2002; Härdle et al, 1993; Xia, 2006; Wang et al, 2010; Cui et al, 2011; Ma & Zhu, 2013). Recently, Jiang & Wang (2010) developed functional single index models for functional/longitudinal response data and derived their associated estimation method and asymptotic theory.…”

Section: Introductionmentioning

confidence: 99%

A Functional Varying-Coefficient Single-Index Model for Functional Response Data

Huang

Zhub

2017

Journal of the American Statistical Association

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Motivated by the analysis of imaging data, we propose a novel functional varying-coefficient single index model (FVCSIM) to carry out the regression analysis of functional response data on a set of covariates of interest. FVCSIM represents a new extension of varying-coefficient single index models for scalar responses collected from cross-sectional and longitudinal studies. An efficient estimation procedure is developed to iteratively estimate varying coefficient functions, link functions, index parameter vectors, and the covariance function of individual functions. We systematically examine the asymptotic properties of all estimators including the weak convergence of the estimated varying coefficient functions, the asymptotic distribution of the estimated index parameter vectors, and the uniform convergence rate of the estimated covariance function and their spectrum. Simulation studies are carried out to assess the finite-sample performance of the proposed procedure. We apply FVCSIM to investigating the development of white matter diffusivities along the corpus callosum skeleton obtained from Alzheimer’s Disease Neuroimaging Initiative (ADNI) study.

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Section: Introductionmentioning

confidence: 99%

A Functional Varying-Coefficient Single-Index Model for Functional Response Data

Huang

Zhub

2017

Journal of the American Statistical Association

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“…Applying Lemma A.4 in Wang et al (2010) and Cauchy-Schwarz inequality, we have, as (log n) 2 /(nh 2 ) → 0,…”

Section: Thus Rmentioning

confidence: 97%

“…Xia and Härdle (2006) applied the minimum average variance estimation (MAVE, Xia et al, 2002) to PLSIM and developed an effective algorithm. More recently, Wang et al (2010) studied estimation in PLSIM with the additional assumptions imposed on model structure. Liang et al (2010) proposed a profile least squares (PrLS) estimation procedure.…”

Section: Introductionmentioning

confidence: 99%

A dimension reduction based approach for estimation and variable selection in partially linear single-index models with high-dimensional covariates

Zhang¹,

Wang²,

Zhu³

et al. 2012

Electron. J. Statist.

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In this paper, we formulate the partially linear single-index models as bi-index dimension reduction models for the purpose of identifying significant covariates in both the linear part and the single-index part through only one combined index in a dimension reduction approach. This is different from all existing dimension reduction methods in the literature, which in general identify two basis directions in the subspace spanned by the parameter vectors of interest, rather than the two parameter vectors themselves. This approach makes the identification and the subsequent estimation and variable selection easier than existing methods for multi-index models. When the number of parameters diverges with the sample size, we then adopt coordinate-independent sparse estimation procedure to select significant covariates and estimate the corresponding parameters. The resulting sparse dimension reduction estimators are shown to be consistent and asymptotically normal with the oracle property. Simulations are conducted to evaluate the performance of the proposed method, and a real data set is analysed for an illustration.

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“…We then apply an iterative algorithm to minimize the objective function with respect to one parameter vector and fixing the others. The iterative algorithm has been commonly used for estimation in partially linear single-index models (PLSiMs) and it converges well (Carroll et al, 1997; Lu and Cheng, 2007; Wang et al, 2010; Xia and Härdle, 2006; Xia et al, 1999). When the nonparametric functions are given, estimation of the parameters follows the same procedure as given in Xia et al (1999).…”

Section: Estimationmentioning

confidence: 99%

SiAM: A hybrid of single index models and additive models

Ma¹,

Lian²,

Liang³

et al. 2017

Electron. J. Statist.

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While popular, single index models and additive models have potential limitations, a fact that leads us to propose SiAM, a novel hybrid combination of these two models. We first address model identifiability under general assumptions. The result is of independent interest. We then develop an estimation procedure by using splines to approximate unknown functions and establish the asymptotic properties of the resulting estimators. Furthermore, we suggest a two-step procedure for establishing confidence bands for the nonparametric additive functions. This procedure enables us to make global inferences. Numerical experiments indicate that SiAM works well with finite sample sizes, and are especially robust to model structures. That is, when the model reduces to either single-index or additive scenario, the estimation and inference results are comparable to those based on the true model, while when the model is misspecified, the superiority of our method can be very great.

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Estimation for a partial-linear single-index model

Cited by 171 publications

References 39 publications

A Functional Varying-Coefficient Single-Index Model for Functional Response Data

A Functional Varying-Coefficient Single-Index Model for Functional Response Data

A dimension reduction based approach for estimation and variable selection in partially linear single-index models with high-dimensional covariates

SiAM: A hybrid of single index models and additive models

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