# Poisson-Gamma Distribution (Negative-Binomial)

### Description

The function fits a mixed Poisson distribution, in which the random parameter follows Gamma distribution (the negative-binomial distribution). As the method of estimation Expectation-maximization algorithm is used. In M-step the analytical formulas taken from [Karlis, 2005] are applied.

### Usage

 1 pg.dist(variable, alpha.start, beta.start, epsylon) 

### Arguments

 variable The count variable. alpha.start The starting value of the parameter alpha. Default to 1. beta.start The starting value of the parameter beta. Default to 0.3 epsylon Default to epsylon = 10^(-8)

### Details

This function provides estimated parameters of the model N|λ \sim Poisson(λ) where λ parameter is also a random variable follows Gamma distribution with hiperparameters α, β. The pdf of Gamma is of the form f_λ(λ)=\frac{λ^{α-1}\exp(-βλ)β^λ}{Γ(α)} .

### Value

 alpha the parameter of mixing Gamma distribution beta the parameter of mixing Gamma distribution theta the value 1/beta n.iter the number of steps in EM algorithm

### References

Karlis, D. (2005). EM algorithm for mixed Poisson and other discrete distributions. Astin bulletin, 35(01), 3-24.

### Examples

 1 2 3 library(MASS) pGamma1 = pg.dist(variable=quine\$Days) print(pGamma1) 

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