A generalization of the negative binomial distribution

Consider a sequence of independent Bernoulli trials with p denoting the probability of success at each trial. With this definition, the probability that the nth success proceed by r failures follows the negative binomial distribution (NB). NB model has been derived from two different forms. At first, the NB can be thought as a Poisson-gamma mixture. The second form of the NB can be derived as a full member of a single parameter exponential family distribution, and therefore considered as a GLM (generalized linear models).
Methods & Materials: We have described a new generalized NB (GNB) distribution with three parameters α, β and k obtained as a compound form of the generalized Poisson and gamma distributions. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. The most important feature of this model is, its time dependent probabilities, and also it can be used for a variety of researches especially in the survival analysis.
This model has been illustrated with two datasets that are indirect measures of illness, along comparing the results of the fitting with NB. Results indicate too much satisfaction. Expected frequencies have been calculated for these data sets to show that the distribution provides a very satisfactory fit in different situations.
Using GNB models allows analyzing very complex data. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. With k = 0 the model becomes the ordinary NB and with α = 1, it becomes a new model which we call it the generalized geometric distribution with two parameters. The most important feature of this model is its time-dependent probabilities.