Extended Geometric Processes: Semiparametric Estimation and Application to Reliability

Lam (2007) introduces a generalization of renewal processes named Geometric processes, where inter-arrival times are independent and identically distributed up to a multiplicative scale parameter, in a geometric fashion. We here envision a more general scaling, not necessarily geometric. The corresponding counting process is named Extended Geometric Process (EGP). Semiparametric estimates are provided and studied for an EGP, which includes consistency results and convergence rates. In a reliability context, arrivals of an EGP may stand for successive failure times of a system submitted to imperfect repairs. In this context, we study: 1) the mean number of failures on some finite horizon time; 2) a replacement policy assessed through a cost function on an infinite horizon time.