Submitted to the Annals of Applied Statistics arXiv: arXiv:0000.0000 A NOTE ON NONPARAMETRIC ESTIMATES OF SPACE-TIME HAWKES POINT PROCESS MODELS FOR EARTHQUAKE OCCURRENCES By Eric Warren Fox ∗ , Frederic Paik Schoenberg ∗ and Joshua Seth Gordon ∗ University of California, Los Angeles ∗ Space-time Hawkes point process models for the conditional rate of earthquake occurrences traditionally make many parametric as- sumptions about the form of the triggering function for the rate of aftershocks following an earthquake. Marsan and Lenglin´ e (2008) de- veloped a completely nonparametric method that provides an esti- mate of a stationary 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 Lenglin´ e’s method in the following ways: (a) the proposal of novel ways to in- corporate a nonstationary background rate; (b) adding error bars to the histogram estimates which capture the sampling variability and bias 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. 1. Introduction. Hawkes point process models of earthquake seismicity usually rely heavily on parametric assumptions about the triggering function for the spatial-temporal rate of aftershock activity following an earthquake. Some important examples are the para- metric forms of the Epidemic Type Aftershock Sequences (ETAS) model of Ogata (1998). Marsan and Lenglin´ e (2008) proposed a more flexible nonparametric approach for estimat- ing Hawkes process models of seismicity which makes no a-priori assumptions about the shape of the triggering function, and provides a data-driven estimate instead. Their method, named Model Independent Stochastic Declustering (MISD), is an iterative algorithm that alternates between first estimating the probability each earthquake in the catalog is either a mainshock or aftershock and second, updating a stationary background rate for mainshock activity and a probability weighted histogram estimate for the triggering function. Nonparametric methods for estimating point process models have shown a wide rage Keywords and phrases: Point processes, nonparametric estimation, Hawkes process, MISD, ETAS model, earthquake forecasting. 1