A nonparametric monotone maximum likelihood estimator of time trend for repairable systems data

The trend-renewal-process (TRP)is defined to be a time-transformed renewal process, where the time transformation is given by a trend function λ(·) which is similar to the intensity of a nonhomogeneous Poisson process (NHPP). A nonparametric maximum likelihood estimator of the trend function of a TRP is obtained under the often natural condition that λ(·) is monotone. An algorithm for computing the estimate is suggested and examined in detail in the case where the renewal distribution of the TRP is is a Weibull distribution. In the case where one has data from several systems, another monotone estimator of λ(·) is suggested, based on the assumption that the superposition of several TRP’s can be approximated by an NHPP.