To make clear what I want to ask, I want to begin with the negative binomial distribution. The first two moments are $E(y)=\mu$ with variance $Var(y)=\mu+\mu^2/k$. When we solve the scorefunction of the log-likelihood with an iterationalgorithm, we get an estimator for $\widehat{\mu}$ (because of the estimator of $\widehat{\beta}$) and an estimator for the shape parameter $\widehat{k}$. So there are two parameters to be estimated.
When I look at the quasi-Poisson approach we have mean $E(y)=\mu$ and variance $Var(y)=\phi\cdot\mu$. This approach is not solvable since we do not know the probability function of $y$. Thus we use quasi-ML methods.
My question is: Why don't we create a distribution where we have the two moments of the quasi-Poisson regression and use a classic ML-estimation? It should be solvable with classic ML-methods since an estimation with negbin distribution (where we have also two estimators for $\mu$ and $k$ compared to $\mu$ and $\phi$).
I'm curious about your answers!