The function poisLindRE is similar to the poisLind
function, but it includes an additional argument group_var that
specifies the grouping variable for the random effects. The function
estimates a Random Effects Poisson-Lindley regression model using
maximum likelihood. It is similar to poisLind, but includes
additional terms to account for the random effects.
The Random Effects Poisson-Lindley model is useful for panel data and
assumes that the random effects follow a gamma distribution. The PDF is
$$
f(y_{it}|\mu_{it},\theta)=\frac{\theta^2}{\theta+1}
\prod_{t=1}^{n_i}\frac{\left(\mu_{it}\frac{\theta(\theta+1)}
{\theta+2}\right)^{y_{it}}}{y_{it}!}
\cdot
\frac{
\left(\sum_{t=1}^{n_i}y_{it}\right)!
\left(\sum_{t=1}^{n_i}\mu_{it}\frac{\theta(\theta+1)}{\theta+2}
+ \theta + \sum_{t=1}^{n_i}y_{it} + 1\right)
}{
\left(\sum_{t=1}^{n_i}\mu_{it}\frac{\theta(\theta+1)}{\theta+2}
+ \theta\right)^{\sum_{t=1}^{n_i}y_{it}+2}
}
$$
The log-likelihood function is:
$$
LL = 2\log(\theta) - \log(\theta+1)
+ \sum_{t=1}^{n_i} y_{it}\log(\mu_{it})
+ \sum_{t=1}^{n_i} y_{it}\log\!\left(
\frac{\theta(\theta+1)}{\theta+2}
\right)
- \sum_{t=1}^{n_i}\log(y_{it}!)
+ \log\!\left(
\left(\sum_{t=1}^{n_i}y_{it}\right)!
\right)
+ \log\!\left(
\sum_{t=1}^{n_i}\mu_{it}\frac{\theta(\theta+1)}{\theta+2}
+ \theta + \sum_{t=1}^{n_i}y_{it} + 1
\right)
- \left(\sum_{t=1}^{n_i}y_{it} + 2\right)
\log\!\left(
\sum_{t=1}^{n_i}\mu_{it}\frac{\theta(\theta+1)}{\theta+2}
+ \theta
\right)
$$
The mean and variance are:
$$\mu_{it}=\exp(X_{it} \beta)$$
$$
V(\mu_{it})=\mu_{it}+
\left(1-\frac{2}{(\theta+2)^2}\right)\mu_{it}^2
$$