I'm solving a problem where I have 15 samples of a unknown distribution. They ask me to fit somehow the parameters of the Binomial, Poisson and Normal distributions. I could use the sample mean and standard deviation to obtain the parameters in the cases of the Normal and Poisson distributions. That's pretty clear. But I'm puzzled with the binomial. The MLE for p assumes that the size N is known, which is not my case. Any idea on what I'm missing here? Can I estimate the size of a unknown binomial distribution?
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Assume $X_1, ...,X_{15} \sim B(n,\pi)$. We know that $E(X) = n\pi$ and $V(X)=n\pi(1-\pi)$.
You can use moment method to get the estimate of the estimate of $n$ and $\pi$. At first get the sample mean $\bar X$ and sample variance $\hat V(X)$, then solve following you will get the estimate.
$$\bar X = \hat n \hat\pi$$ $$\hat V(X) = \hat n \hat\pi(1-\hat \pi)$$
answered Jul 19, 2019 at 17:29
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$\begingroup$ Ok, just to make it clear. This problem requires using the moment method. There's no closed solution with maximum likelihood alone. Right? $\endgroup$ Commented Jul 19, 2019 at 17:41
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$\begingroup$ Of course you can try MLE. But maybe more complicated. $\endgroup$ Commented Jul 19, 2019 at 17:43