An exponential regression-based model with stochastic intensity is developed to describe the software reliability growth phenomena, where the software testing metrics depend on the intensity process. For such a generalized modeling framework, the common maximum likelihood method cannot be applied any more to the parameter estimation. In this paper, we propose to use the pseudo maximum likelihood method for the parameter estimation and to seek not only the model parameters but also the software reliability measures approximately. It is shown in numerical experiments with real software fault data that the resulting software reliability models based on four parametric approximations provide the better goodness-of-fit performance than the common non-homogeneous Poisson process models without testing metric information.