The Product Relative Error Criteria Based Model Analysis

Positive response appears commonly in many practical problems,such as econom-ic filed or survival analysis.To handle the positive response,we generally consider the multiplicative regression model,which is known as the accelerated failure time(AFT)model in survival analysis when the response variable is a failure time.For the posi-tive response variable,the relative errors,rather than the absolute errors,are the major concern.In this article,based on the least product relative error(LPRE)criterion,we consider the multiplicative regression model with error-in-covariate or right censor-ing.In addition,we extend the LPRE criterion to the varying coefficient multiplicative regression model to capture the underling complex relationship between the response variable and their associated covariables.In Chapter 2,based on LPRE criterion,we propose two estimating equation ap-proaches to estimate the regression parametric vector in the multiplicative regression model when a subset of covariates are subject to measurement error but replicate mea-surements of their surrogates are available.The first method is to construct an unbiased estimating equation based on conditional mean score,whereas the second method is to correct the naive method to obtain an unbiased estimating equation.Both methods allow the number of replicate measurements to vary between subjects.No parametric assumption is imposed on the measurement error term and the true covariates which are not observed in the data set.Under some regularity conditions,the asymptotic nor-mality is proved for both the proposed estimators.Furthermore,a theoretical compari-son is made for them in a special case where the distribution of the measurement error follows the normal distribution.Some simulation studies are conducted to assess the performances of the proposed methods.A real data analysis on the ACTG315 datasets(Lederman et al.,1998)is used to illustrate our methods.In Chapter 3,the multiplicative regression model with right censoring is consid-ered.This model is useful for modeling the survival data in survival analysis.Under the random right censorship,the LPRE criterion was extended to the case with censor-ing.Under some regular conditions,the consistency and asymptotic normality were established.Some simulations were conducted to examine the finite performance of the proposed method.Finally,as an illustration,we applied the proposed method to the nwtco datasets(D'angio et al.,1989;Green et al.,1998).In Chapter 4,we study the censored multiplicative regression model with covari-ates measured with error.When there is no censoring,this model will be the multi-plicative regression model with error-in-covariates.When the covariates are measured precisely,this model is the censored multiplicative regression model.Assume that repli-cate measurements of the error-prone covariates are available.Based on the LPRE cri-terion,we propose two objective functions by inverse censoring probability weighting techniques.The two estimator of the parameters is given by minimizing the two objec-tive functions respectively.Using a combination of martingale and estimating equation techniques,we show that two proposed estimators for the regression coefficients are consistent and asymptotically normal.Simulation studies are conducted to examine the performance of the two proposed estimators.For illustration,our two proposed methods are applied to analyse a data set arising from a clinical trial(Fuchs et al.1994).In Chapter 5,we aim to extend the LPRE criterion to the varying coefficient mul-tiplicative regression model and propose the local least product relative error criterion for estimating the coefficient functions.This extension involves nonparametric estima-tion and kernel smoothing techniques.The objective function of the proposed criterion has the advantage of scale free,which is appealing in the analysis of survival data and financial data.The consistency and asymptotic normality of the proposed estimation are established.Some numerical simulations are carried out to assess the performance of the proposed estimator.As an illustration,the ethanol data(Brinkman,1981)is an-alyzed.