In the count data set, the frequency of some points may occur more than expected under the standard data analysis models. Indeed, in many situations, the frequencies of zero and of some other points tend to be higher than those of the Poisson. Adapting existing models for analyzing inflated observations has been studied in the literature. A method for modeling the inflated data is the inflated distribution. In this paper, we extend this inflated distribution}. Indeed, if inflations occur in three or more of the support point, then the previous models are not suitable. We propose a model based on zero, one, ..., and k inflated points with probabilities w_0,w_1,..., and w_k, respectively. By choosing the appropriate values for the weights w_0,...,w_k, various inflated distributions, such as the zero-inflated, zero-one-inflated, and zero-k-inflated distributions, are derived as special cases of the proposed model in this paper. Various illustrative examples and real data sets are analyzed using the obtained results.

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