Estimating historic movement of a climatological variable from a pair of misaligned functional data sets

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.