Let's start with the Poisson process:
If $N_t$ is a Poisson process with parameter $\lambda$, then we know that the inter-arrival time distribution is an exponential distribution with parameter $\lambda$. As such, the parameter $\lambda$ can be interpreted as the intensity of arrivals. Moreover, $\mathbb{E}[N_t] = \lambda t$. For the annual intensity, we can then write,
\begin{equation} \mathbb{P}(N=n) = \frac{e^{\lambda}\lambda^n}{n !} \end{equation} It is possible to find an MLE estimation of $\lambda$ for a given observation.
Now suppose that $N_t$ is a renewal process for which the inter-arrival times are distributed as gamma with parameters $\eta_1$ and $\eta_2$. Applying the Laplace transform method, one can find the $\mathbb{E}[N_t]$, which is a function of parameters $\eta_1$ and $\eta_2$. It is feasible to derive the probability function of $N_t$ in case $\eta_1$ is a positive integer. That is \begin{equation} \mathbb{P}(N_t = n) = \sum_{i= n\eta_1}^{n\eta_1 + \eta_1 -1}\frac{e^{-\eta_2}\eta_2^{i}}{i!} \end{equation} I wonder if we can find the estimation of parameters $\eta_1$ and $\eta_2$ by using MLE through the above probability function or through the inter-arrival time distribution (which is gamma)?