Nonparametric estimation in trend-renewal processes

•To construct an algorithm to estimate the components of a trend renewal process.•To extend previous approaches for which the renewal hazard is on parametric form.•To evaluate the finite sample properties through simulations.•To apply the new method to real data sets.

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(·)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 can be obtained in principle in a similar manner as for the NHPP using kernel smoothing. For a full nonparametric estimation of a trend-renewal process it is necessary, however, to estimate jointly the trend function and the renewal distribution. For this purpose we consider a nonparametric approach using kernel smoothing techniques. We develop an original algorithm to estimate the conditional intensity function by preserving its structure in terms of the trend function and the underlying renewal process. The algorithm is applied to both simulated and real data sets.