  Data depth is a well-known and useful nonparametric tool for analyzing
functional data. It provides a novel way of ranking a sample of curves from the
center outwards and defining robust statistics, such as the median or trimmed
means. It has also been used as a building block for functional outlier
detection methods and classification. Several notions of depth for functional
data were introduced in the literature in the last few decades. These
functional depths can only be directly applied to samples of curves measured on
a fine and common grid. In practice, this is not always the case, and curves
are often observed at sparse and subject dependent grids. In these scenarios
the usual approach consists in estimating the trajectories on a common dense
grid, and using the estimates in the depth analysis. This approach ignores the
uncertainty associated with the curves estimation step. Our goal is to extend
the notion of depth so that it takes into account this uncertainty. Using both
functional estimates and their associated confidence intervals, we propose a
new method that allows the curve estimation uncertainty to be incorporated into
the depth analysis. We describe the new approach using the modified band depth
although any other functional depth could be used. The performance of the
proposed methodology is illustrated using simulated curves in different
settings where we control the degree of sparsity. Also a real data set
consisting of female medflies egg-laying trajectories is considered. The
results show the benefits of using uncertainty when computing depth for sparse
functional data.
