Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

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 non-normal correlated random errors. However, the proposed estimating functions are non-smooth and non-convex. 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 smooth-threshold estimating equations is developed to simultaneously identify non-zero 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 method.

Penalized weighted smoothed quantile regression for high‐dimensional longitudinal data

“…with C 1 defined in (18), and c = c 0 𝜅 * f l . Proposition S3 shows that the event  rsc (a, b, c) holds with probability at least 1 − (2p…”

“…Fu and Wang, 15 Tang and Leng, 16 and Leng and Zhang 17 all indicated that the efficiency of the parameter estimation will be lost when a strong correlation exists. To capture the correlation and enhance the estimation efficiency, Lv et al 18 used the smooth‐threshold efficient QR‐based generalized estimating equations to select the covariates and estimate the parameters in the quantile partially linear additive model with longitudinal data; Wang and Sun 19 explored this kind of method for the quantile partial linear varying coefficient model with longitudinal data.…”

Variable Selection for Additive Models with Missing Response at Random

“…Based on the centered spline basis function approximation, we propose two new imputed estimating equation methods to estimate the component functions using the smooth-threshold estimating equation proposed in Ueki [9]. The variable selection method based on the smooth-threshold estimating equation can avoid the convex optimization of the penalized variable selection procedure, and has been extended to some models, such as Li et al [10], Zhao and Li [11], Lv et al [12] and Geronimia and Saportab [13]. In the present paper, based on the marginal imputed estimating equation and the maximum correlation imputed estimating equation, we define two smooth-threshold estimating equations to study model (1) with missing response at random.…”