Negative binomial regression includes special cases among which geometric regression can be referred to.In geometric regression, the dispersion coefficient is taken as one. Geometric regression is as same as theregular multiple regression, but it differs from it in that the dependent variable (Y) is in this case an observed number that follows a geometric distribution.In this case, the non-negative integers, ie 0, 1, 2, etc., represent possible values of the dependent variableY. Generalized Poisson regression (GPR) is used to estimate the regression coefficients and to deal with cases of over-dispersion and under-dispersion, where the geometric regression is a case of generalized Poisson regression (GPR) in which the assumption that the variance is equal to the mean can be overcome as stated in the Poisson model. Geometric regression has many applications that this study aims to explain. A set of variables (x's) called regression variables and exposure time called t can be used to calculate the mean y in the geometric regression. We can express the mentioned quantities through the following equation: 𝜇𝑖 = 𝑒xp(ln(𝑡𝑖) + 𝛽1𝑥1𝑖 + 𝛽2𝑥2𝑖 + ⋯+ 𝛽𝑘𝑥ki) In many cases, β1 is called the intercept when 𝑥1 ≡ 1, and the regression coefficients are expressed as β1, β2,…, βk which is a set of unknown parameters that can be determinedusing a set of data, and their estimatedquanities are denoted by b1, b2, ..., bk. Summing up from the above, if we assume that we have an observation i, then in this case we can write the basic geometric regression model with the following statement: Pr (Y=yi|μi) = Γ(yi+1)/Γ(yi+1) (1/(1+μi))^1 〖(μi/(1+μi))〗^yi Using the method of maximum likelihood, we are able to estimate the regression coefficients. As a generalization of Poisson's regression, geometric regression is used for modelling the count data. Modeling count data is a common task in economics and the social sciences.modeling count data can be employed in many applications in the economic, social and other sciences. The family of generalized linear models includes many models that can be used in modeling counting data and employing its applications in many fields. Some of the models that belong to this family are geometric regression, classical Poisson model, and negative binomial regression. The R System for Statistical Computing has a toolbox containing these models (Achim et al., 2019). However, Classical count data models (Poisson, NegBin) often not fexible enough for applications in economics and the social sciences. Since empirical counting data sets typically show an increased scattering and/or an excessive number of zeros, In such cases, the Classic Poisson model is usually unable to provide models for counting data.
