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The estimation of branching point process models by maximum likelihood can be unstable and computationally intensive. We explore an alternative estimation method based on the Expectation-Maximization algorithm. The method involves viewing the estimation of such branching processes as analogous to incomplete data problems. Using an application from seismology, we show how the Epidemic-type Aftershock Sequence (ETAS) model can in fact be estimated this way and we propose a computationally efficient procedure to maximize the log-likelihood function. Using a space-time ETAS model, we demonstrate that this method is extremely robust and accurate and use it to estimate declustered background seismicity rates of geologically distinct regions in Southern California. All regions show similar declustered background intensity estimates except for the one covering the Southern section of the San Andreas fault system to the East of San Diego in which a substantially higher intensity is observed.

This work presents an alternative derivation of the asymptotic distribution of Ripley's K-function for a homogeneous Poisson process and shows how it can be combined with point process residual analysis in order to test for different classes of point process models. This can be doen with the mean K-function of thinned residuals (KM ) or a weighted analogue called the weighted or inhomogeneous K-function (KW ). The asymptotic distributions of KM and KW are derived for inhomogeneous Poisson processes. Both statistics can be used as measures of goodness-of-fit for point process models.

We investigate the properties of a weighted analogue of Ripley’s K-function which was first introduced by Baddeley, Møller, and Waagepetersen. This sta- tistic, called the weighted or inhomogeneous K-function, is useful for assessing the fit of point process models. The advantage of this measure of goodness- of-fit is that it can be used in situations where the null hypothesis is not a stationary Poisson model. We note a correspondence between the weighted K- function and thinned residuals, and derive the asymptotic distribution of the weighted K-function for a spatial inhomogeneous Poisson process. We then present an application of the use of the weighted K-function to assess the goodness-of-fit of a class of point process models for the spatial distribution of earthquakes in Southern California.