On semiparametric inference for modulated renewal processes

Modulated renewal processes, suggested by D. R. Cox, give a flexible way to introduce dependencies into point processes. We discuss the asymptotics of partial likelihood inference for modulated renewal processes when the random covariate for the process involves its history. We show, in some generality, that the estimators of the regression parameter and the cumulative hazard have the same asymptotic distributions that they would have under the usual proportional hazards model, even though the martingale justification for partial likelihood no longer applies because of a reordering of the time-scale. An example is given to illustrate the ideas. A simulation study is presented to confirm the theoretical results.