On the use and misuse of time-rescaling to assess the goodness-of-fit of self-exciting temporal point processes

The paper first highlights important drawbacks and biases related to the common use of time-rescaling to assess the goodness-of-fit (Gof) of self-exciting temporal point process (SETPP) models. Then it presents a new predictive time-rescaling approach leading to an asymptotically unbiased Gof framework for general SETPPs in the case of single observed trajectories. The predictive approach focuses on forecasting accuracy and addresses the bias problem resulting from the plugged-in estimated parameters. Dawid's prequential approach is used and the models' checking is mainly based on the forecasting accuracy of arrival times. These times are transformed, using sequentially estimated parameters, into random vectors which are proved to converge in probability under the null hypothesis and standard regulatory conditions to vectors of iid Exponential(1) rv's. Numerical experiments are used to compare the performances of the standard and predictive time-rescaling for Gof assessment of non-homogeneous Poisson and Hawkes self-exciting temporal processes. Data of Japanese seismic events are also used to illustrate the dynamic aspect of the proposed model-checking approach.