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)