Estimate $m$ using method of maximum likelihood

Question

Estimate $m$ using method of maximum likelihood.

In the box there are $91$ balls, where $m$ are red, and the rest are blue. To estimate unknown parameter $m$, at once $19$ balls are drawn, $7$ being red and $12$ being blue. Based on the given sample, estimate $m$ using method of maximum likelihood.

Im aware this is the hypogeometric distribution in question. Im having trouble finding the maximum using MLE and how would I use these $19$ drawn balls to get the answer?

Answer 1

1. Start with a simple-minded approach.

The likelihood and log-likelihood functions are very easy to evaluate. Up to a scaling constant on the likelihood (which doesn't alter where the maximum is), the likelihood or log-likelihood functions can easily be plotted:

(Left panel is scaled likelihood, the right panel shifted log-likelihood, focused in on a region near the maximum)

[From there the argmax can be identified by inspection; if you want to prove it's the maximum, it's relatively easy algebraically once you identify its approximate location, which is already obvious from the plot. On larger problems, there are various approximations (binomial, Poisson, normal) that can be useful in getting an approximate region to look in]

2. Once you're used to simply manipulating and plotting the likelihood on a couple of problems,

   (a) try to see if you can show that the ratio of successive likelihoods ($\mathcal{L}(m)/\mathcal{L}(m-1)$) is $>1$ for small values of $m$ and $<1$ for large values of $m$, and

   (b) that it's monotonic.

   If you can then identify the conditions under which it exceeds 1, deriving the MLE should be straightforward.