A comparison of some random effect models for parameter estimation in recurrent events

We consider maximum likelihood estimation for multiplicative intensity models with random effects arising from point processes. The methods of estimation include Gauss-Hermite integration based on log-normal random effects and the EM algorithm for nonparametric estimation of the mixing distribution. The former approximates the marginal likelihood by the Gauss-Hermite rule and the latter is most suitable for discrete random effects. We contrast these two methods of estimation with respect to the bias, relative efficiency, and coverage probability of the parameter estimates. We demonstrate, via simulation, that the regression parameter estimates from these two methods have negligible bias and their variance estimates are also valid for practical use. This desirable feature is also robust to misspecification of the mixing. The estimate for the variance parameter under the log-normal random effect model may have small positive bias if the true mixing distribution is highly discrete. In contrast, the EM algorithm for a nonparametric random effect distribution provides practically unbiased estimate for the variance though on occasion it will give an unrealistically large value. We provide empirical evidence that specification of the baseline intensity function as a piecewise constant function is quite robust to misspecification of the baseline intensity function.