Maximum Likelihood Estimation for Zero-inflated Poisson distribution I am trying to do exactly what the title says. What I have is the log-likelihood function as follows:
Likelihood function, where $I_i = 1$ when $X_i = 0$, and $I_i = 0$ otherwise. Then I took the partial derivatives of that like this. I tried simplifying this expression to get an equation, but I keep getting nonsense. After searching the web, I found this book but I don't understand how the author arrived at ML equations just from looking at the PGF, can someone help explain that please? There is also this entry but they use a different model from mine (p = probability of Poisson, 1-p = probability of 0, whereas both the first book and my model use p = probability of 0, 1-p = probability of Poisson) and again they don't show the simplification steps so I can't really use that. Thanks for any help.

I think the inconsistency is caused by an error in the derivative of the profile likelihood in Ben's answer: The $\lambda e^{-\lambda}$ in the numerator should only be $ e^{-\lambda}$. In this case the final equation is
$$ \bar{x}(1-e^{-\hat{\lambda}})=\hat{\lambda}(1-r_0)(1-e^{\hat{\lambda }}+e^{\hat{\lambda }})=\hat{\lambda}(1-r_0),$$
which is the solution from the cited book. This equation can be either solved iteratively or directly by using the main branch of Lambert's W function as shown in the reference given below: Defining $\gamma=\frac{\bar{x}}{1-r_0}$ the solution is $$\hat{\lambda}=W_0(-\gamma e^{-\gamma})+\gamma$$
See Dencks et al.:Assessing Vessel Reconstruction in Ultrasound Localization Microscopy by Maximum Likelihood Estimation of a Zero-Inflated Poisson Model. DOI:10.1109/TUFFC.2020.2980063
