Unobserved heterogeneity in the power law nonhomogeneous Poisson process

A study of possible consequences of heterogeneity in the failure intensity of repairable systems is presented. The basic model studied is the nonhomogeneous Poisson process with power law intensity function. When several similar systems are under observation, the assumption that the corresponding processes are independent and identically distributed is often questionable. In practice there may be an unobserved heterogeneity among the systems. The heterogeneity is modeled by introduction of unobserved gamma distributed frailties. The relevant likelihood function is derived, and maximum likelihood estimation is illustrated. In a simulation study we then compare results when using a power law model without taking into account heterogeneity, with the corresponding results obtained when the heterogeneity is accounted for. A motivating data example is also given. •Consequences of overlooking heterogeneity in similar repairable systems are studied.•Likelihood functions are established for power law NHPP w/ and w/o heterogeneity.•ML estimators for parameters of power law NHPP with heterogeneity are derived.•A simulation study shows the effects of heterogeneity and its ignorance in models.