In this article, we formulate a semiparametric model for counting processes in which the effect of covariates is to transform the time scale for a baseline rate function. We assume an arbitrary dependence structure for the counting process and propose a class of estimating equations for the regression parameters. Asymptotic results for these estimators are derived. In addition, goodness of fit methods for assessing the adequacy of the accelerated rates model are proposed. The finite-sample behavior of the proposed methods is examined in simulation studies, and data from a chronic granulomatous disease study are used to illustrate the methodology.