Asymptotics of minimum distance estimation for self-exciting load sharing point processes

Abstract The cumulative intensity of a self-exciting point process allows us to translate complex dynamics such as load sharing effects and the accumulation of damage into a parametric model. Kopperschmidt and Stute (Stat Sin 23:1273–1298, 2013) proposed a minimum Cramér–von Mises distance estimator for semi-parametric cumulative intensity models. However, their proof of the asymptotic normality of the estimator is based on an invalid extension of Kolmogorov’s tightness criterion to arbitrary dimensions. In this paper, we correct the proof of the asymptotic normality of the minimum distance estimator for self-exciting point processes. We show that our slightly stronger assumptions are satisfied for load sharing processes with damage accumulation, an important class of self-exciting point processes in engineering sciences. Moreover, Wald-type confidence regions can be derived from the asymptotic normal distribution of the minimum distance estimator. We apply the corresponding tests to a real data set to confirm that the effects of load sharing and damage accumulation are statistically significant. We also compare them to the likelihood-ratio test in a simulation study, and assess the robustness of the minimum distance estimator claimed by Kopperschmidt and Stute (2013) by applying both methods to a contaminated data set.