Smoothing combined generalized estimating equation

Authors: Jing Lv Hu Yang Chaohui Guo

Publish Date: 2015/08/12

Volume: 31, Issue: 3, Pages: 1203-1234

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Abstract

This paper develops a robust and efficient estimation procedure for quantile partially linear additive models with longitudinal data where the nonparametric components are approximated by B spline basis functions The proposed approach can incorporate the correlation structure between repeated measures to improve estimation efficiency Moreover the new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations method for nonnormal correlated random errors However the proposed estimating functions are nonsmooth and nonconvex In order to reduce computational burdens we apply the induced smoothing method for fast and accurate computation of the parameter estimates and its asymptotic covariance Under some regularity conditions we establish the asymptotically normal distribution of the estimators for the parametric components and the convergence rate of the estimators for the nonparametric functions Furthermore a variable selection procedure based on smooththreshold estimating equations is developed to simultaneously identify nonzero parametric and nonparametric components Finally simulation studies have been conducted to evaluate the finite sample performance of the proposed method and a real data example is analyzed to illustrate the application of the proposed methodWe are grateful for the insightful comments from the anonymous reviewers and editors which have greatly helped improve the quality of this paper This work is supported by the Chongqing University Postgraduates’ Innovation Project the National Natural Science Foundation of China Grant No 11171361 and PhD Programs Foundation of Ministry of Education of China Grant No 20110191110033Eg lz l = 0 and g l in mathscr H r l=1ldots d for some r1/2 where mathscr H r is the collection of all functions on 0 1 whose rho th order derivative satisfies the Hddotolder condition of the order v with r equiv rho + v and 0vle 1There exist constants delta 1 and delta 2 such that the marginal density f lz l of Z l satisfies 0 delta 1 le f lz l le delta 2 infty on 0 1 for every l=1ldots d The joint density f llz lz l of Z lZ l satisfies 0 delta 1 le f llz lz lle delta 2 infty for all z lz lin 012 1le lne lle dThe dimensions p d of varvecx ij and varvecz ij are fixed and max n i is bounded when mrightarrow infty The distribution functions F ijt=p y ij varvecx ijTvarvecbeta sum nolimits l = 1d g lz ijl le tleft varvecx ijvarvecz ij right are absolutely continuous with continuous densities f ij uniformly bounded and its first derivative f ij cdot uniformly bounded away from 0 and infty at the points 0 i=1ldots m j=1ldots n iFor any positive definite matrix varvecW i mathrm m 1sum nolimits i = 1m varvecH iTvarvecLambda ivarvecW i varvecLambda ivarvecH i converges to a positive definite matrix where varvecLambda i is an n i times n i diagonal matrix with the jth diagonal element f ij0 and sup iVert varvecH iVert + infty Let lambda max=mathrmmaxlambda 1lambda 2 and lambda min=mathrmminlambda 1lambda 2 The tuning parameters lambda min and lambda max satisfy mr/2r+1lambda max rightarrow 0 and m1+kappa r/2r+1lambda min rightarrow infty as m rightarrow infty

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