Abstract: We consider a semi-parametric self-exciting point process regression model where the excitation function is assumed to be smooth and decreasing but otherwise unspecified, and the baseline intensity is assumed to be a linear function of the regressors. We propose an estimation method for this model based on monotone splines. The estimator for the regression coefficients is shown to be consistent, asymptotically normal, and semi-parametrically efficient. The consistency of the estimator for the nonparametric excitation function is also established. The numerical performance of the estimators was found to be satisfactory through simulation studies. We illustrate the application of the model to a bladder cancer data set.

Key words and phrases: B-spline, efficient estimator, Hawkes process, monotone spline, point process, self-exciting process, semiparametric efficiency, sieve estimator.