Let $Y_1,\dots,Y_n$ be indepedendent and Poisson random variables with $E(Y_i)=\exp(\alpha+\beta z_i)$. Find the MLE of $\theta=(\alpha,\beta)$ and its asymptotic distribution.
Question: It is not easy to find the MLE directly. It seems that it is a hierarchical model.
For the asymptotic distribution. It seems that the delta method may be applied.
Can anyone give some hint? Especially, how to calculate the MLE?
I thought $Y|\bigwedge$~$Poisson (\bigwedge)$
$\bigwedge$~$Exp(\alpha+\beta Z)$
Thus,
$f_Y(y)=\int_0^{+\infty} \frac{z^y e^{-z}}{y!} e^{(\alpha+\beta z)}dz|$
Though I can not make sure whether $Exp(\alpha+\beta Z)$ is a distribution
and how to calculate $f_Y(y)$Many thanks!