A class of birth processes having a variety of practical applications in penetration of new services and products is considered. Typically, statistical inferences on these models are performed by means of simple error structures placed on the deterministic analogs of the underlying stochastic processes. Motivated by the poor performance of conventional estimation methods, the problem of estimating the parameters of these models is readdressed. We develop necessary formulae for performing the maximum likelihood estimation and weighted least squares estimation methods, and demonstrate their superiority through analyses of some real data and simulation studies.