  The notion of statistical depth has been extensively studied in multivariate
and functional data over the past few decades. In contrast, the depth on
temporal point process is still under-explored. The problem is challenging
because a point process has two types of randomness: 1) the number of events in
a process, and 2) the distribution of these events. Recent studies proposed
depths in a weighted product of two terms, describing the above two types of
randomness, respectively. In this paper, we propose to unify these two
randomnesses under one framework by a smoothing procedure. Basically, we
transform the point process observations into functions using conventional
kernel smoothing methods, and then adopt the well-known functional $h$-depth
and its modified, center-based, version to describe the center-outward rank in
the original data. To do so, we define a proper metric on the point processes
with smoothed functions. We then propose an efficient algorithm to estimated
the defined "center". We further explore the mathematical properties of the
newly defined depths and study asymptotics. Simulation results show that the
proposed depths can properly rank the point process observations. Finally, we
demonstrate the new method in a classification task using a real neuronal spike
train dataset.
