Christine H. Müller

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.

We consider two models for estimating the expected states of nodes in networks where the observations at nodes are given by random states and measurement errors. In the first model, we assume independent successive observations at the nodes and the design question is how often the nodes should be observed to obtain a precise estimation of the expected states. In the second model, all nodes are observed simultaneously and the design question is to determine the nodes which need larger precision of the measurements than other nodes. Both models lead to the same design problem. We derive explicitly A-optimal designs for the most simple network with star configuration. Moreover, we consider the network with wheel configuration and derive some conditions which simplify the numerical calculation of the corresponding A-optimal designs.

The recently proposed simple but powerful sign depth tests depend on the order of the residuals. While one-dimensional explanatory variables provide a natural order, there exists no canonical order for multidimensional explanatory variables. For this scenario, we present different approaches for ordering multidimensional explanatory variables and compare them regarding their performance with respect to the stability of the ordering, the usability for non-metric explanatory variables, the computational time complexity, and in the context of testing in linear models including high-dimensional multiple regression. It is shown that the sign depth tests based on orderings given by pairwise distances perform best. They are much more powerful than the classical sign test and also than the F-test when not having normally distributed errors. They are competitive to the much more complicated robust Wald test based on efficient MM-estimation. Additionally, the sign depth tests are more appropriate for outlier robust model checks.

To model damage accumulation for load-sharing systems, two models given by intensity functions of self-exciting point processes are proposed: a model with additive damage accumulation and a model with multiplicative damage accumulation. Both models include the model without damage accumulation as a special case. For both models, the likelihood functions are derived and maximum likelihood estimators and likelihood ratio tests are given in a scale-invariant version and a scale-dependent version. Furthermore, a Bayesian approach using Markov chain Monte Carlo methods for posterior computation is provided. The frequentist and Bayesian methods are applied to a data set of failures of tension wires of concrete beams where a significant damage accumulation effect is confirmed by both additive and multiplicative damage accumulation models. This is all the more remarkable as a simulation study indicates that the tests for an existing damage accumulation effect are rather conservative. Moreover, prediction intervals for the failure times of the tension wires in a new experiment are given, which improve former prediction intervals derived without damage accumulation. The simulation study considers a scenario with a fixed time horizon and one with fixed numbers of failed components of the systems.