Year of Publication

2018

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Statistics

First Advisor

Dr. Richard Charnigo

Abstract

Methods of estimating parametric and nonparametric components, as well as properties of the corresponding estimators, have been examined in partially linear models by Wahba [1987], Green et al. [1985], Engle et al. [1986], Speckman [1988], Hu et al. [2004], Charnigo et al. [2015] among others. These models are appealing due to their flexibility and wide range of practical applications including the electricity usage study by Engle et al. [1986], gum disease study by Speckman [1988], etc., wherea parametric component explains linear trends and a nonparametric part captures nonlinear relationships.

The compound estimator (Charnigo et al. [2015]) has been used to estimate the nonparametric component of such a model with multiple covariates, in conjunction with linear mixed modeling for the parametric component. These authors showed, under a strict orthogonality condition, that parametric and nonparametric component estimators could achieve what appear to be (nearly) optimal rates, even in the presence of subject-specific random effects.

We continue with research on partially linear models with subject-specific random intercepts. Inspired by Speckman [1988], we propose estimators of both parametric and nonparametric components of a partially linear model, where consistency is achievable under an orthogonality condition. We also examine a scenario without orthogonality to find that bias could still exist asymptotically. The random intercepts accommodate analysis of individuals on whom repeated measures are taken. We illustrate our estimators in a biomedical case study and assess their finite-sample performance in simulation studies.

Jump points have often been found within the domain of nonparametric models (Muller [1992], Loader [1996] and Gijbels et al. [1999]), which may lead to a poor fit when falsely assuming the underlying mean response is continuous. We study a specific type of change-point where the underlying mean response is continuous on both left and right sides of the change-point. We identify the convergence rate of the estimator proposed in Liu [2017] and illustrate the result in simulation studies.

Digital Object Identifier (DOI)

https://doi.org/10.13023/ETD.2018.201

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