In many longitudinal settings, time-varying covariates may not be measured at the same time as responses and are often prone to measurement error. Naive last-observation-carried-forward methods incur estimation biases, and existing kernel-based methods suffer from slow convergence rates and large variations. To address these challenges, we propose a new functional calibration approach to efficiently learn longitudinal covariate processes based on sparse functional data with measurement error. Our approach, stemming from functional principal component analysis, calibrates the unobserved synchronized covariate values from the observed asynchronous and error-prone covariate values, and is broadly applicable to asynchronous longitudinal regression with time-invariant or time-varying coefficients. For regression with time-invariant coefficients, our estimator is asymptotically unbiased, root-n consistent, and asymptotically normal; for time-varying coefficient models, our estimator has the optimal varying coefficient model convergence rate with inflated asymptotic variance from the calibration. In both cases, our estimators present asymptotic properties superior to the existing methods. The feasibility and usability of the proposed methods are verified by simulations and an application to the Study of Women's Health Across the Nation, a large-scale multi-site longitudinal study on women's health during mid-life.

翻译：在许多纵向研究中，时变协变量与响应变量的测量时间往往不同步，且常伴有测量误差。传统的末次观测值结转法会导致估计偏差，而现有的基于核函数的方法则面临收敛速度慢、方差大的问题。为应对这些挑战，我们提出一种新型函数校准方法，基于含测量误差的稀疏函数数据高效学习纵向协变量过程。该方法源于函数主成分分析，能从观测到的异步含误差协变量值中校准出未观测到的同步协变量值，并广泛适用于时不变或时变系数的异步纵向回归。对于时不变系数回归，估计量渐近无偏、根号n相合且渐近正态；对于时变系数模型，估计量在最优变系数模型收敛速度下因校准导致渐近方差增大。两种情形下，估计量的渐近性质均优于现有方法。通过模拟实验和"全国女性健康研究"（一项关于中年女性健康的大规模多中心纵向研究）的应用，验证了所提方法的可行性和实用性。