Temporal point processes (TPPs) model the timing of discrete events along a timeline and are widely used in fields such as neuroscience and fi- nance. Statistical depth functions are powerful tools for analyzing centrality and ranking in multivariate and functional data, yet existing depth notions for TPPs remain limited. In this paper, we propose a novel product depth specifically designed for TPPs observed only up to the first k events. Our depth function comprises two key components: a normalized marginal depth, which captures the temporal distribution of the final event, and a conditional depth, which characterizes the joint distribution of the preceding events. We establish its key theoretical properties and demonstrate its practical utility through simulation studies and real data applications.
@article{arxiv.2511.19375,
title = {Product Depth for Temporal Point Processes Observed Only Up to the First k Events},
author = {Chifeng Shen and Yuejiao Fu and Xiaoping Shi and Michael Chen},
journal= {arXiv preprint arXiv:2511.19375},
year = {2025}
}