#### Date of Award

2018

#### Document Type

Open Access Dissertation

#### Department

Statistics

#### First Advisor

Xianzheng Huang

#### Abstract

The error-in-covariates problem has received great attention among researchers who study semiparametric and nonparametric inference for regression models over the past two decades. Without correcting for the measurement error in covariates, estimators for covariate eﬀect usually contain bias. To account for measurement error, much research have been done in mean regression (Liang et al., 1999; Fuller, 2009; Carroll et al., 2006) and quantile regression (He and Liang, 2000; Hardle et al., 2000; Wei and Carroll, 2009). In contrast, there is little research in mode regression and this motivates us to propose semiparametric methods to address this error-incovariates problem in Chapters 1 and 3. Chapter 1 considers estimating the mode of a response given an error-prone covariate X by assuming that the mode of Y given X is a linear function of X. It is frst shown that ignoring measurement error typically leads to inconsistent inference for the mode of the response given the true covariate, as well as misleading inference for regression coefcients in the conditional mode model. To account for measurement error, the Monte Carlo corrected score method (Novick and Stefanski, 2002) is employed to numerically obtain an unbiased score function based on which the regression coefcients is estimated consistently. To relax the normality assumption on measurement error the frst method requires, the corrected kernel method is proposed. In this method, an objective function constructed using deconvoluting kernels is maximized to obtain consistent estimators of the regression coefcients. Besides rigorous investigation on large sample properties of the new estimators, we study their fnite sample performance via extensive simulation experiments, and fnd iv that the proposed methods substantially outperform a naive inference method that ignores measurement error. In Chapter 2, we assume that the mode of Y is a linear function of a covariate X and it also depends on another covariate T in an unspecifed functional form. This leads to a partially linear model for the conditional mode. We employ B-splines to approximate the unspecifed function that relates Y and T. To estimate the covariate eﬀects explaining the association between Y and X, and at the same time, estimate the unspecifed function linking Y and T, we develop two methods for inferring these two parts of the partially linear mode model. A simulation study is designed to show the performance of two proposed methods. Chapter 3 considers estimating the mode of a response in partially linear models when the aforementioned X is error-prone. To account for measurement error, we incorporate the corrected kernel method proposed in Chapter 1 and the proposed estimation methods in Chapter 2 to infer the parametric part and nonparametric part of the conditional mode accounting for measurement error in X. Results from simulation studies suggest that the proposed method substantially outperform a naive inference method that ignores measurement error. Instead of considering error-prone covariates, in Chapter 4, we consider a scenario where the response is contaminated by Berkson measurement error. In particular, we tackle the regression analysis for a pooled continuous response. Finally, Chapter 5 discusses future research in my dissertation

#### Recommended Citation

Li, X.(2018). *Semiparametric Regression In The Presence Of Measurement Error.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/4937