This paper presents a semiparametric regression approach for estimating the mean and rate functions of recurrent events. The method is based on a counting process model with a Cox-type intensity function, which assumes that the underlying counting process is a time-transformed Poisson process. However, the authors propose a more robust approach that does not rely on the Poisson-type assumption. They use modern empirical process theory to justify this robust procedure and develop methods for constructing simultaneous confidence bands for the mean function and checking the adequacy of the fitted models. The advantages of the robust procedures are demonstrated through simulation studies, and an application to multiple-infection data from a clinical study on chronic granulomatous disease is provided. The paper introduces a semiparametric model for recurrent events that allows for arbitrary dependence structures among events. It defines the mean function as the expected number of events over time and the rate function as the instantaneous rate of events. The model is shown to be more flexible than the traditional Cox model, as it does not assume independence of events. The authors derive the asymptotic distribution of the regression parameters under this model and develop methods for estimating the mean function and its variance. They also propose graphical and numerical techniques for checking the adequacy of the fitted models. The paper also discusses the use of robust variance estimators and shows that they provide more accurate estimates of the true variance of the regression parameters compared to naive estimators. The authors conduct simulation studies to evaluate the performance of the robust method and compare it to the traditional Cox model. The results show that the robust method performs better, especially in the presence of overdispersion. The paper concludes with a discussion of the implications of the results for the analysis of recurrent events. It highlights the importance of using robust methods in the presence of complex dependence structures and the need for further research on the properties of the proposed models. The authors also note that the results can be extended to other rate models and that the methods can be applied to a wide range of problems involving recurrent events.This paper presents a semiparametric regression approach for estimating the mean and rate functions of recurrent events. The method is based on a counting process model with a Cox-type intensity function, which assumes that the underlying counting process is a time-transformed Poisson process. However, the authors propose a more robust approach that does not rely on the Poisson-type assumption. They use modern empirical process theory to justify this robust procedure and develop methods for constructing simultaneous confidence bands for the mean function and checking the adequacy of the fitted models. The advantages of the robust procedures are demonstrated through simulation studies, and an application to multiple-infection data from a clinical study on chronic granulomatous disease is provided. The paper introduces a semiparametric model for recurrent events that allows for arbitrary dependence structures among events. It defines the mean function as the expected number of events over time and the rate function as the instantaneous rate of events. The model is shown to be more flexible than the traditional Cox model, as it does not assume independence of events. The authors derive the asymptotic distribution of the regression parameters under this model and develop methods for estimating the mean function and its variance. They also propose graphical and numerical techniques for checking the adequacy of the fitted models. The paper also discusses the use of robust variance estimators and shows that they provide more accurate estimates of the true variance of the regression parameters compared to naive estimators. The authors conduct simulation studies to evaluate the performance of the robust method and compare it to the traditional Cox model. The results show that the robust method performs better, especially in the presence of overdispersion. The paper concludes with a discussion of the implications of the results for the analysis of recurrent events. It highlights the importance of using robust methods in the presence of complex dependence structures and the need for further research on the properties of the proposed models. The authors also note that the results can be extended to other rate models and that the methods can be applied to a wide range of problems involving recurrent events.