The problem and some of my thought are as followed, could you help check if I'm wrong.
Suppose $X\sim\mathrm{Bin} (n_1,1/2)$ and $Z\sim\mathrm{Bin}(n_2,p)$, $0<p<1$ being an unknown parameter; $X$ and $Z$ are assumed to be independent. Due to (spatial) aggregation, we can only observe $Y=X+Z$.
Is there always an MLE of $p$?
With the replicate of $Y$, and by the convolution formula of pdf, $$f_Y(y)=\sum_{x=0}^nf_X(x)f_Z(y-x)=\sum_{x=0}^nC_n^x(1/2)^n\times C_n^x p^x(1-p)^{n_2-x}$$ something like this. And then, we construct the likelihood function, find its argmax solve for $p$. I think it is kind of obvious, only the calculation is annoying.
But what does the question really mean? Is there any case that a MLE will not exist?
And are there consistent estimators of $p$ based on $Y$ alone?