This paper addresses the analysis of recurrent event data using semiparametric regression models. The authors provide a rigorous justification for robust procedures that relax the Poisson-type assumption, using modern empirical process theory. They propose methods for constructing simultaneous confidence bands for the mean function and develop graphical and numerical techniques to assess the adequacy of fitted models. The advantages of these robust procedures are demonstrated through simulation studies and an application to multiple-infection data from a clinical trial on chronic granulomatous disease (CGD). The paper also discusses the limitations of intensity-based models and highlights the flexibility and efficiency of the proposed rate and mean models in handling complex dependence structures.This paper addresses the analysis of recurrent event data using semiparametric regression models. The authors provide a rigorous justification for robust procedures that relax the Poisson-type assumption, using modern empirical process theory. They propose methods for constructing simultaneous confidence bands for the mean function and develop graphical and numerical techniques to assess the adequacy of fitted models. The advantages of these robust procedures are demonstrated through simulation studies and an application to multiple-infection data from a clinical trial on chronic granulomatous disease (CGD). The paper also discusses the limitations of intensity-based models and highlights the flexibility and efficiency of the proposed rate and mean models in handling complex dependence structures.