A non-parametric monotone maximum likelihood estimator of time trend for repairable system 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 image which is similar to the intensity of a non-homogeneous Poisson process (NHPP). A non-parametric maximum likelihood estimator of the trend function of a TRP is obtained under the often natural condition that image 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 a Weibull distribution. The case where one has data from several systems is also briefly studied.