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**Spatially inhomogeneous background rate estimators and uncertainty quantification for nonparametric Hawkes point process models of earthquake occurrences.**
*(English)*
Zbl 1391.62271

Summary: Space-time Hawkes point process models for the conditional rate of earthquake occurrences traditionally make many parametric assumptions about the form of the triggering function for the rate of aftershocks following an earthquake. As an alternative, D. Marsan and O. Lengline [“Extending earthquakes’ reach through cascading”, Science 319, No. 5866, 1076–1079 (2008; doi:10.1126/science.1148783)] developed a completely nonparametric method that provides an estimate of a homogeneous background rate for mainshocks, and a histogram estimate of the triggering function. At each step of the procedure the model estimates rely on computing the probability each earthquake is a mainshock or aftershock of a previous event. The focus of this paper is the improvement and assessment of Marsan and Lengliné’s method in the following ways: (a) the proposal of novel ways to incorporate a spatially inhomogeneous background rate; (b) adding error bars to the histogram estimates which quantify the sampling variability in the estimation of the underlying seismic process. A simulation study is designed to evaluate and validate the ability of our methods to recover the triggering function and spatially varying background rate. An application to earthquake data from the Tohoku District in Japan is discussed at the end, and the results are compared to a well-established parametric model of seismicity for this region.

### MSC:

62P12 | Applications of statistics to environmental and related topics |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

62M30 | Inference from spatial processes |