Profile local linear estimation of generalized sem

Authors: Yanqing Sun Liuquan Sun Jie Zhou

Publish Date: 2013/03/08

Volume: 19, Issue: 3, Pages: 317-349

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Abstract

This paper studies the generalized semiparametric regression model for longitudinal data where the covariate effects are constant for some and timevarying for others Different link functions can be used to allow more flexible modelling of longitudinal data The nonparametric components of the model are estimated using a local linear estimating equation and the parametric components are estimated through a profile estimating function The method automatically adjusts for heterogeneity of sampling times allowing the sampling strategy to depend on the past sampling history as well as possibly timedependent covariates without specifically model such dependence A Kfold crossvalidation bandwidth selection is proposed as a working tool for locating an appropriate bandwidth A criteria for selecting the link function is proposed to provide better fit of the data Large sample properties of the proposed estimators are investigated Large sample pointwise and simultaneous confidence intervals for the regression coefficients are constructed Formal hypothesis testing procedures are proposed to check for the covariate effects and whether the effects are timevarying A simulation study is conducted to examine the finite sample performances of the proposed estimation and hypothesis testing procedures The methods are illustrated with a data exampleThe authors thank the reviewers for their constructive comments that have improved the presentation and content of the paper The research of Yanqing Sun was partially supported by NSF grants DMS0905777 and DMS1208978 NIH NIAID grant 2 R37 AI05416510 and a fund provided by UNC Charlotte The research of Liuquan Sun was partly supported by the National Natural Science Foundation of China Grants No 10731010 10971015 and 10721101 the National Basic Research Program of China 973 Program No 2007CB814902 and Key Laboratory of RCSDS CAS No 2008DP173182The covariate processes X icdot and Z icdot are left continuous The censoring time C i is noninformative in the sense that EdN itX itZ itC ige t=EdN itX itZ it and EY itX itZ itC ige t=EY itX itZ it dN it is independent of Y it conditional on X itZ it and C ige t the processes Y itX itZ it and alpha it0le tle tau are bounded and their total variations are bounded by a constant EN it 2N it 12le Lt 2t 1 for 0le t 1le t 2le tau where L0 is a constant the link function gy is monotone and its inverse function g1x is twice differentiable gamma 0te xxt and e xzt are twice differentiable e xxt1 is bounded over 0le tle tau the matrices A and Sigma are positive definite the weight process Wtxzmathop longrightarrow limits Pwtxz uniformly in the range of txz wtxz is differentiable with uniformly bounded partial derivatives the kernel function Kcdot is symmetric with compact support on 11 and bounded variation bandwidth hrightarrow 0 EN it+hN ith2+v=Oh for some v 0 the limit lim nrightarrow infty hEint 0tau w isY ismu isX is K hst d N isotimes 2= Sigma et exists and is finiteLet u agamma beta =Evarphi gamma 0TtX it+beta 0TZ itvarphi gamma Tt X it+beta TZ it X itxi italpha it Define gamma beta t as the unique root such that u agamma beta beta =0 for beta in mathcalN beta where mathcalN beta is a neighborhood of beta 0 Let e beta xxt=Ebig w itdotvarphi gamma beta TtX it+beta TZ it X itotimes 2 alpha itxi it big and e beta xzt=Ebig w itdotvarphi gamma beta TtX it+beta TZ it X it Z itTalpha itxi it big When beta =beta 0 we have gamma beta t=gamma 0t In this case e beta xxt=e xxt and e beta xzt=e xzt Let gamma abeta t=gamma T beta tmathbf0 qTT where mathbf0 q is a qtimes 1 vector of zerosLet H=mathrmdiagI qhI q The following lemmas are used in the proofs of the main theorems The proofs of the lemmas make repeated applications of the GlivenkoCantelli Theorem Theorem 194 of van der Vaart 1998 A sufficient condition for applying the GlivenkoCantelli Theorem can be checked by estimating the order of the bracketing number similar to the proof of Lemma 2 of Sun et al 2009 This sufficient condition holds under the conditions provided in Condition A The details are omitted to save spaceBy a similar argument Hpartial 2 tildegamma tbeta /partial beta 2 converges in probability to a deterministic function of tbeta of bounded variation uniformly in tin t 1t 2 and beta in mathcalN beta square

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