Extended Geometric Processes: Semiparametric Estimation and Application to ReliabilityImperfect repair, Markov renewal equation, replacement policy

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 necessar- ily 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 suc- cessive failure times of a system submitted to imperfect repairs. In this context, we study: 1) the mean number of failures on some finite hori- zon time 2) a replacement policy assessed through a cost function on an infinite horizon time.