# Poisson-Log_normal Distribution

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### Description

The function fits a mixed Poisson distribution, in which the random parameter follows Log_normal distribution. As the method of estimation Expectation-maximization algorithm is used. In M-step the GLM is applied.

### Usage

 1 pln.dist.glm(variable, lambda.start, nu.start, epsylon, n) 

### Arguments

 variable The count dependent variable in the regression lambda.start The starting value of lambda parameter of Poisson distribution. Default to 1. nu.start The starting value of delta parameter of Log_normal distribution. Default to 1. epsylon Default to epsylon = 10^(-8) n The integer value for the Laguerre quadrature. Default to 100.

### Details

This function provides estimated parameters of the model N|θ \sim Poisson(λ θ) where θ is a latent variable comes from log-normal distribution with one parameter ν. The pdf of log-normal is of the form f_θ(θ)=\frac{1}{√{2πνθ}}\exp[-\frac{(\log(θ)+\frac{ν^2}{2})^2}{2ν^2}] . The parameter λ is determined by the intercep β_0 through log-link λ=\exp(β_0).

### Value

 lambda fixed effect in mixed Poisson distribution nu the parameter of mixing log-normal distribution n.iter n likelihood.values values of log-likelihood

### References

Ghitany, M. E., Karlis, D., Al-Mutairi, D. K., & Al-Awadhi, F. A. (2012). An EM algorithm for multivariate mixed Poisson regression models and its application. Applied Mathematical Sciences, 6(137), 6843-6856.

### Examples

 1 2 3 4 library(MASS) var = quine$Days[quine$Days<11] plogn = pln.dist.glm(variable=var) print(plogn) 

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