2
$\begingroup$

Consider the Urn problem where a hypothetical urn contains a finite number of $m$ balls, $r$ of which are black and $m-r$ are white. We take a random sample without replacement of $n$ balls and observe $k$ are black. The distribution of $K$ follows the hypergeometric distribution with mass $$P(K=k) = \frac{{n \choose k}{m-r \choose n-k}}{{m \choose n}}$$

I am interested in the "best" integer approximation of $r$ given known $m,n$ and observed $k$.

It can be shown that $$\hat{r} = \frac{m}{n}{k}$$ is an unbiased, consistent estimator of $r$. However, it can take non-integer values and is thus an "implausible" point estimate of $r$ for a finite population. I weary of simply rounding $\hat{r}$ to the nearest integer value given the asymmetry of the confidence intervals are such an estimate.

Is there a recommended method for estimating $$\tilde{r}\approx r,\qquad \tilde{r}\in\mathbb{Z}^+$$ Is it reasonable to choose $\tilde{r}\in\{\lfloor \hat{r}\rfloor,\lceil \hat{r} \rceil\}$ such that $$P(K=k|m=m,n=n,r=\tilde{r})$$is maximized?

$\endgroup$
3
$\begingroup$

Basically

$$ P(K=k|m=m,n=n,r=\tilde{r}) $$

is the likelihood function, so you could as well conduct a grid search among any integers within the valid values for $r$ to maximize the likelihood. If you start with the rounded values returned by the estimator, you are just bounding the search.

However if you look at the A Note About Maximum Likelihood Estimator in Hypergeometric Distribution paper by Hanwen Zhang, you'll see that $\lfloor \frac{m}{n}k\rfloor$ is the MLE estimator, so the procedure is not needed for non-integers.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.