Abstract
In the analysis of point processes or recurrent events, the self-exciting component can be an important factor in understanding and predicting event occurrence. A Cox-type self-exciting intensity point process is generally not a proper model because of its explosion in finite time. However, the model with $m$-memory is appropriate to analyze sequences of recurrent events. It assumes the most recent $m$ events multiplicatively affect the conditional intensity of event occurrence. Aside from the interpretability, one advantage is the simplicity of the estimation and inference--the Cox partial likelihood can be applied and the resulting estimator is consistent and asymptotically normal. Another advantage is that the model can be applied to the analysis of case-cohort data via the pseudo-likelihood approach. The simulation studies support the asymptotic theory. Application is illustrated with analysis of a bladder cancer dataset and of an Australian stock index dataset, which shows evidence of self-excitation.
Highlights
Recurrent event data are encountered frequently in many areas of scientific endeavor, such as the modeling and predictions of earthquakes and other disastrous events, study of the patterns of neural firings in neuroscience, assessing the efficacy of cancer medications in suppressing the recurrence of tumors, and analysis of the risk of default on debt repayments by borrowers
In the analysis of point processes or recurrent events, the self-exciting component can be an important factor in understanding and predicting event occurrence
The estimated covariate effects are clearly inflated and the standard errors are generally underestimated. This implies that application of the Cox proportional intensities (CoxPI) model to recurrent event data without accounting for the potential self-exciting effect may lead to erroneous inference about the covariate effects
Summary
Recurrent event data are encountered frequently in many areas of scientific endeavor, such as the modeling and predictions of earthquakes and other disastrous events, study of the patterns of neural firings in neuroscience, assessing the efficacy of cancer medications in suppressing the recurrence of tumors, and analysis of the risk of default on debt repayments by borrowers. If the data is in the form of a single long string of event recurrence times, it might be of interest to predict the event recurrence time by exploiting potential dependence of the waiting times between events on past events or on exogenous covariates Models of this type include the self-exciting point process (Hawkes, 1971; Ogata, 1978), the modulated renewal process (Cox, 1972; Oakes & Cui, 1994; Lin & Fine, 2009) and the autoregressive conditional duration models (Engle & Russell, 1998; Fernandes & Grammig, 2006).
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