  The problem of estimating missing fragments of curves from a functional
sample has been widely considered in the literature. However, a majority of the
reconstruction methods rely on estimating the covariance matrix or the
components of its eigendecomposition, a task that may be difficult. In
particular, the accuracy of the estimation might be affected by the complexity
of the covariance function and the poor availability of complete functional
data. We introduce a non-parametric alternative based on a novel concept of
depth for partially observed functional data. Our simulations point out that
the available methods are unbeatable when the covariance function is
stationary, and there is a large proportion of complete data. However, our
approach was superior when considering non-stationary covariance functions or
when the proportion of complete functions is scarce. Moreover, even in the most
severe case of having all the functions incomplete, our method performs well
meanwhile the competitors are unable. The methodology is illustrated with two
real data sets: the Spanish daily temperatures observed in different weather
stations and the age-specific mortality by prefectures in Japan.
