Given RV’s $Y_1,\ldots,Y_n$ all of which i.i.d. with Poisson distribution with fixed $\lambda$. The question is to show that the MLE based on only the number of $Y_i$’s that are zero, say $n_0$, is equal to $\ln(n)-\ln(n_0)$.
I’ve made several attempts to show this, but I honestly don’t have the time to put all of it into MathJax, so I’ll describe what I did, and the problems I encountered. I tried maximizing the following expression w.r.t. $\lambda$
$$ \prod_{i=1}^n f_{Y_i}(y_i \mid n_0), $$ but don’t even know if this is the correct expression to try to maximize. I took the logarithm and split the sum up into two pieces, namely the terms where $y_i$ was $0$, and where it wasn’t. Then I substituted in probabilities and took the derivative eventually leading to $\lambda = \bar{Y}_n$, which is just the normal MLE. I don’t know what I might’ve done wrong.
If I may be so frank to request a kind soul out there to provide me with a clear answer, that’d be awesome! My exam is tomorrow, and I feel like I’m missing something pretty fundamental here. I’d also like to mention that I’d like to provide much more context, and don’t like this question myself, but I’m really short for time. Thanks in advance and thanks for understanding.
The question is essentially asking for an MLE for $\lambda$ in the situation where the only information you can observe is the number of $Y_i$s that are equal to zero. You do not have access to the actual values of the $Y_i$, so you cannot make use of $y_i$ except in counting $y_i = 0$ events.
If that's the case, then what we really have is $n$ independent indicator variables $I_i = \mathbb{I}(Y_i = 0)$. From that:
Can you identify $P(I_i = 1)$?
Can you identify what distribution $n_0 = \sum I_i$ should have?
Can you then construct an MLE based on that?