Nonhomogeneous Poisson process with nonparametric frailty

The failure processes of heterogeneous repairable systems are often modeled by non-homogeneous Poisson processes. The common way to describe an unobserved heterogeneity between systems is to multiply the basic rate of occurrence of failures by a random variable (a so-called frailty) having a specified parametric distribution. Since the frailty is unobservable, the choice of its distribution is a problematic part of using these models, as are often the numerical computations needed in the estimation of these models. The main purpose of this paper is to develop a method for estimation of the parameters of a nonhomogeneous Poisson process with unobserved heterogeneity which does not require parametric assumptions about the heterogeneity and which avoids the frequently encountered numerical problems associated with the standard models for unobserved heterogeneity. The introduced method is illustrated on an example involving the power law process, and is compared to the standard gamma frailty model and to the classical model without unobserved heterogeneity. The derived results are confirmed in a simulation study which also reveals several not commonly known properties of the gamma frailty model and the classical model, and on a real life example.