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：在许多纵深环境中,时间差异的共变点可能无法与应对措施同时测量,而且往往容易发生测量错误。最后观察和偏差的逆向方法产生估计偏差,而现有的内核方法存在缓慢趋同率和巨大差异。为了应对这些挑战,我们提议一种新的功能校准方法,以高效学习基于零位功能数据、测量误差的纵向共变进程。我们的方法来自功能性主要构件分析,它校准了观察到的不同步和易误差的共变值中未观测到的同步共变值,并且广泛适用于带有时间变化或时间变化系数的无同步长度回归。对于与时间差异系数的回归,我们的估测器是偏偏重、根一致和不中性正常;对于时间变化系数模型,我们的估测器有最佳的相异系数模型趋同率,与校正性差相膨胀。在这两种情况下,我们目前对女性健康状况进行大规模模拟研究时,其目前采用的是高性的方法。