Abstract
The focus of this study is efficient estimation in a quantile regression model with partially linear coefficients for longitudinal data, where repeated measurements within each subject are likely to be correlated. We propose a weighted quantile regression approach for time-invariant and time-varying coefficient estimation. The proposed approach can employ two types of weights obtained from an empirical likelihood method to account for the within-subject correlation: the global weight using all observations and the local weight using observations in the neighborhood of the time point of interest. We investigate the influence of choice of weights on asymptotic estimation efficiency and find theoretical results that are counter intuitive; it is essential to use the global weight for both time-invariant and time-varying coefficient estimation. This benefits from the within-subject correlation and prevents an adverse effect due to the weight discordance. For statistical inference, a random perturbation approach is utilized and evaluated through simulation studies. The proposed approach is also illustrated through a Multi-Center AIDS Cohort study.
Highlights
Efficient estimation has been attracting significant attention in parametric quantile regression (QR) models for longitudinal data; see [9, 6, 19, 5, 11, 20, 14, 3].this has not been investigated in semiparametric or nonparametricQR models
In response to this gap in literature, this paper thoroughly investigates estimation efficiency in the partially linear quantile regression model; for τ ∈ (0, 1), the τ th conditional quantile of the jth response from subject i measured at time tij, denoted by Yi(tij), is formulated as
To fit the partially linear QR model for longitudinal data, [21] considers B-spline basis functions, yet it does not achieve estimation efficiency due to ignoring the within-subject correlation commonly existing in longitudinal studies
Summary
Efficient estimation has been attracting significant attention in parametric quantile regression (QR) models for longitudinal data; see [9, 6, 19, 5, 11, 20, 14, 3]. To fit the partially linear QR model for longitudinal data, [21] considers B-spline basis functions, yet it does not achieve estimation efficiency due to ignoring the within-subject correlation commonly existing in longitudinal studies. We have investigated the impact of the choice of weights and have obtained very interesting results that oppose to one’s intuition in accommodating the local weight and global weight for efficient estimation of βτ (t) and ατ , respectively. The global weight should be used in both estimation of βτ (t) and ατ to achieve estimation efficiency of ατ ; the amount of estimation efficiency for ατ gained from the proposed approach corresponds to that obtained in [19] under the parametric QR model. Regularity conditions and proofs of theoretical results are provided in the Appendix
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