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 λ ( · ) 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 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 a Weibull distribution. The case where one has data from several systems is also briefly studied.