Most real life count data consists of some values that are more frequent than allowed by the common parametric families of distributions. For data consisting of only excess zeros, in a seminal paper Lambert (1992) introduced Zero-Inflated Poisson (ZIP) model, which is a mixture model that accounts for the inflated zeros. In this thesis, two Doubly Inflated Poisson (DIP) probability models, DIP (p, λ) and DIP ( p1, p2, λ), are discussed for situations where there is another inflated value k \u3e 0 besides the inflated zeros. The distributional properties such as identifiability, moments, and conditional probabilities are also discussed for both probability models. For the data consisting of raw counts as well as grouped frequencies, we have considered parameter estimation using maximum likelihood (ML) and method of moments techniques. Efficiencies show that the ML estimators perform far better than the moment estimators. An application to DIP models is illustrated using data on patients\u27 length of stay in a hospital. Parameter estimation of DIP regression models using maximum likelihood approach is also discussed using data on dental cavities. Finally, we conclude with a brief introduction to two Doubly Inflated Negative Binomial (DINB) distributions and their related regression models
