We consider the problem of estimating the mean function from a pair of paleoclimatic functional data sets after one of them has been registered with the other. We establish that registering one data set with respect to the other is the appropriate way to formulate this problem. This is in contrast with estimation of the mean function on a ‘central’ time scale that is preferred in the analysis of multiple sets of longitudinal growth data. We show that if a consistent estimator of the time transformation is used for registration, the Nadaraya–Watson estimator of the mean function based on the registered data would be consistent under a few additional conditions. We study the potential change in asymptotic mean squared error of the estimator because of the contribution of the time-transformed data set. We demonstrate through simulations that the additional data can lead to improved estimation despite estimation error in registration. Analysis of three pairs of paleoclimatic data sets reveals some interesting points.

我们考虑了在一对古气候功能数据集已彼此注册之后从一对古气候功能数据集估计均值函数的问题。我们确定相对于另一数据集注册一个数据集是解决此问题的适当方法。这与“中央”时间尺度上的均值函数估计相反，后者在分析多组纵向增长数据时首选。我们证明，如果使用时间变换的一致估计量进行配准，则在某些其他条件下，基于注册数据的均值函数的Nadaraya–Watson估计量将保持一致。由于时间转换数据集的作用，我们研究了估计量的渐进均方误差的潜在变化。我们通过仿真证明，尽管注册过程中存在估计错误，但其他数据仍可以导致改进的估计。对三对古气候数据集的分析揭示了一些有趣的观点。