The objective of this paper is to provide an alternative distribution for modeling overdispersed count data. We propose a negative binomial-Crack (NB-CR) distribution which is obtained by mixing the NB distribution with a CR distribution. This new formulation distribution contains as special cases three parameter distribution, namely, negative binomial-inverse Gaussian (NB-IG), negative binomial-Birnbaum-Saunders (NB-BS) and negative binomial-length biased inverse Gaussian (NB-LBIG). In addition, we present some properties of the new distribution such as the factorial moments, the first four moments, variance, skewness and kurtosis. Parameters estimation are also implemented using maximum likelihood method and the application of NB-CR distribution is carried out on a sample of count data. The results show that the NB-CR provides a better fit compared to the Poisson and the NB distribution.

In this paper, a new mixed negative binomial (NB) distribution named as negative binomial-weighted Garima (NB-WG) distribution has been introduced for modeling count data. Two special cases of the formulation distribution including negative binomial- Garima (NB-G) and negative binomial-size biased Garima (NB-SBG) are obtained by setting the specified parameter. Some statistical properties such as the factorial moments, the first four moments, variance and skewness have also been derived. Parameter estimation is implemented using maximum likelihood estimation (MLE) and real data sets are discussed to demonstrate the usefulness and applicability of the proposed distribution.

In this paper, we introduced models for zero inflated and zero truncated based on the negative binomial-weighted Garima (NB-WG) distribution. The zero inflated negative binomialweighted Garima (ZINB-WG) distribution is a discrete probability distribution for the excessive zero counts and overdispersion which is a mixture of Bernoulli distribution and negative binomial-weighted Garima (NB-WG) distribution. Meanwhile, a new zero truncated distribution named as the zero truncated negative binomial-weighted Garima (ZTNB-WG) distribution can be used when the response variable is the set of positive integers. Some properties of the two different versions of NB-WG distribution are discussed and the estimation of the parameters is derived by maximum likelihood method (MLE). In addition, the usefulness of the proposed distributions is illustrated by real data sets.

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