Modeling continuous time series with many zeros and an application to earthquakes

We introduce a way to discretize a point process such as earthquakes into time series and propose a class of two‐part autoregressive (2PAR) models for continuous time series data with excess zeros to analyze the dependence structure in such discretized point processes. In the 2PAR models, we employ a Bernoulli variable to model the excess zeros and use autoregressive processes to describe the serial correlation. Using this class of models, we can model correlations that exist in either zeros or nonzeros in the data. We introduce a class of residual analysis to check the goodness‐of‐fit of the proposed models and a forecasting procedure using simulation to check the performance of the models. A simulation study illustrates that the estimators of the proposed models are consistent. Application to the 2010 Darfield earthquake sequence in New Zealand shows that the 2PAR models with serial correlation in both the presence probability and the earthquake stress release provide higher information gain against a reference model.