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13
Functional generalized additive models
- Journal of Computational and Graphical Statistics
, 2014
"... We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect to t of F{X(t), t} where F (·, ·) is an unknown regression fu ..."
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Cited by 13 (3 self)
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We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect to t of F{X(t), t} where F (·, ·) is an unknown regression function and X(t) is a functional covariate. Rather than having an additive model in a finite number of principal components as in Müller and Yao (2008), our model incorporates the functional predictor directly and thus our model can be viewed as the natural functional extension of generalized additive models. We estimate F (·, ·) using tensor-product B-splines with roughness penalties. A pointwise quantile transformation of the functional predictor is also considered to ensure each tensor-product B-spline has observed data on its support. The methods are evaluated using simulated data and their predictive performance is compared with other competing scalar-on-function regression alternatives. We illustrate the usefulness of our approach through an application to brain tractography, where X(t) is a signal from diffusion tensor imaging at position, t, along a tract in the brain. In one example, the response is disease-status (case or control) and in a second example, it
Partially linear models with missing response variables and error-prone covariates
- Biometrika
, 2007
"... In this paper, we consider partially linear models in the form Y = XTβ + ν(Z) + ε when the response variable Y is sometimes missing with missingness probability pi depending on (X,Z), and the covariate X is measured with error, where ν(z) is an unspecified smooth function. The missingness structure ..."
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Cited by 4 (1 self)
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In this paper, we consider partially linear models in the form Y = XTβ + ν(Z) + ε when the response variable Y is sometimes missing with missingness probability pi depending on (X,Z), and the covariate X is measured with error, where ν(z) is an unspecified smooth function. The missingness structure is therefore missing not at random (NMAR), rather than the usual missing at random (MAR). We propose a class of semiparametric estimators for parameter of interest β, as well as for the population mean E(Y). The resulting estimators are shown to be consistent and asymptotically normal under general assumptions. To construct a confidence region for β, we also propose an empirical likelihood based statistic, which is shown to have an asymptotic chi-squared distribution. The proposed methods are applied to analyze an AIDS clinical trial data set. A simulation study is also reported to illustrate our approach.
Backfitting versus profiling in general criterion functions
- Statist. Sinica
, 2007
"... Abstract: We study the backfitting and profile methods for general criterion func-tions that depend on a parameter of interest β and a nuisance function θ. We show that when different amounts of smoothing are employed for each method to esti-mate the function θ, the two estimation procedures produce ..."
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Cited by 1 (0 self)
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Abstract: We study the backfitting and profile methods for general criterion func-tions that depend on a parameter of interest β and a nuisance function θ. We show that when different amounts of smoothing are employed for each method to esti-mate the function θ, the two estimation procedures produce estimators of β with the same limiting distributions, even when the criterion functions are non-smooth in β and/or θ. The results are applied to a partially linear median regression model and a change point model, both examples of non-smooth criterion functions. Key words and phrases: Backfitting, change points, dioxin, kernel estimation, me-dian regression, nonparametric regression, partially linear model, profile kernel
Functional Generalized Additive Models
"... Disclaimer: This is a version of an unedited manuscript that has been accepted for publication. As a service to authors and researchers we are providing this version of the accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript bef ..."
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Disclaimer: This is a version of an unedited manuscript that has been accepted for publication. As a service to authors and researchers we are providing this version of the accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to this version also. PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use:
Simultaneous variable selection and
"... estimation in semiparametric modeling of longitudinal/clustered data ..."
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High dimensionality Model selection Oracle property
, 2012
"... Shrinkage methods a b s t r a c t We study variable selection for partially linear models when the dimension of covariates diverges with the sample size. We combine the ideas of profiling and adaptive Elastic-Net. The resulting procedure has oracle properties and can handle collinearity well. A by-p ..."
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Shrinkage methods a b s t r a c t We study variable selection for partially linear models when the dimension of covariates diverges with the sample size. We combine the ideas of profiling and adaptive Elastic-Net. The resulting procedure has oracle properties and can handle collinearity well. A by-product is the uniform bound for the absolute difference between the profiled and original predictors. We further examine finite sample performance of the proposed procedure by simulation studies and analysis of a labor-market dataset for an illustration. & 2012 Elsevier B.V. All rights reserved.
SEMIPARAMETRIC MIXTURE OF BINOMIAL REGRESSION WITH A DEGENERATE COMPONENT
"... Abstract: Many datasets contain a large number of zeros, and cannot be modeled directly using a single distribution. Motivated by rain data from a global climate model, we study a semiparametric mixture of binomial regressions, in which both the component proportions and the success probabilities de ..."
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Abstract: Many datasets contain a large number of zeros, and cannot be modeled directly using a single distribution. Motivated by rain data from a global climate model, we study a semiparametric mixture of binomial regressions, in which both the component proportions and the success probabilities depend on the predictors nonparametrically. An EM algorithm is proposed to estimate this semiparametric mixture model by maximizing the local likelihood function. We also consider a special case in which the component proportions are constant while the component success probabilities still depend on the predictors nonparametrically. This model is estimated by a one-step backfitting procedure, and the estimates are shown to achieve the optimal convergence rates. The asymptotic properties of the estimates for both models are established. The proposed procedures are applied to rain data from a global climate model and historical rain data from Edmonton, Canada. Simulation studies show that satisfactory estimates are obtained for the proposed models for finite samples. Key words and phrases: climate change, EM algorithm, weather data 1
Separation of Covariates into Nonparametric and Parametric Parts in High-Dimensional Partially Linear Additive Models
, 2013
"... Determining which covariates enter the linear part of a partially linear additive model is always challenging. This challenge becomes more serious when the number of covari-ates diverges with the sample size. In this paper, we propose a double penalization based procedure to distinguish covariates t ..."
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Determining which covariates enter the linear part of a partially linear additive model is always challenging. This challenge becomes more serious when the number of covari-ates diverges with the sample size. In this paper, we propose a double penalization based procedure to distinguish covariates that enter the nonparametric and parametric parts and to identify insignificant covariates simultaneously for the “large p small n ” setting. The procedure is shown to be consistent for model structure identification. That is, it can iden-tify zero, linear and nonlinear components correctly. Moreover, the resulting estimators of the linear coefficients are shown to be asymptotically normal. We also discuss how to choose the penalty parameters and provide theoretical justification. We conduct extensive simulation experiments to evaluate the numerical performance of the proposed methods and analyze a gene data set for an illustration.
