  We consider in this paper 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 show theoretically that registering one data set
with respect to the other is the right way to formulate this problem, which is
in contrast with estimation of the mean function in a "neutral" time scale that
is preferred in the analysis of multiple sets of longitudinal growth data. Once
this registration is done, the Nadaraya-Watson estimator of the mean function
may be computed from the pooled data. We show that, if a consistent estimator
of the time transformation is used for this registration, the above estimator
of the mean function would be consistent under a few additional conditions. We
study the potential change in asymptotic mean squared error of the estimator
that may be possible because of the contribution of the time-transformed data
set. After demonstrating through simulation that the additional data can lead
to improved estimation in spite of estimation error in registration, we
estimate the mean function of three pairs of paleoclimatic data sets. The
analysis reveals some interesting aspects of the data sets and the estimation
problem.
