Nonhomogeneous Poisson process with nonparametric frailty and covariates

The nonhomogeneous Poisson process is commonly used in the modeling of failure times of complex repairable systems. In practice there may be a substantial heterogeneity in the failure behavior among apparently identical repairable systems. In this paper we introduce a new approach for statistical modeling of failures and the corresponding statistical inference when there is both an observable and unobservable heterogeneity between such systems. The approach is partly nonparametric and hence avoids making restrictive assumptions about the underlying process. The main feature of the approach is the elimination of the effect of unobservable heterogeneity, which leaves an optimization problem involving the observable covariates only. The new method is introduced in a power law process setting and can easily be extended to general nonhomogeneous Poisson process. The satisfactory performance of the method is verified in an extensive simulation study as well as in a case study, and the method compares favorably to the gamma frailty model and to the classical regression model not assuming an unobserved heterogeneity. The approach can be adapted for a wide class of models.