Estimator for E[A]/E[B]^2 I have 2 random variables $A$ and $B$, with unknown means (denoted $\overline A$ and $\overline B$), and I take $n$ samples $\{<A_i,B_i>|i=1..n\}$. If I want to estimate $\overline A/\overline B^2$, $\frac{\sum A_i/n}{(\sum B_i/n)^2}$ is biased. Is there an unbiased estimator (or a bias correction) for this?
If $A$ is the sum of two elements of a vector random variable with a Dirichlet-Multinomial distribution, and $B$ is the difference of the elements, does that affect the estimator I should use?
 A: Partial answer:
You can derive some bias corrections for this following a similar approach to the one used for $\bar A/\bar B$ in Myint Tin (Journal of the American Statistical Association, Vol. 60, No. 309 (Mar., 1965), pp. 294-307).
First express the biassed estimator (using $\bar a =\sum A_i /n$) as
$\bar a/ \bar b^2=\frac{\bar A(1+\delta \bar a)}{\bar B^2(1+\delta \bar B)^2}$
where  $\delta \bar a=\frac{\bar a-\bar A}{\bar A}$ and $\delta \bar b=\frac{\bar b-\bar B}{\bar B}$. Then assume that $|\delta \bar b|<1$ so that you can expand $\frac{1}{(1+\delta \bar b)^2}$ as a Taylor expansion: $\frac{1}{(1+\delta \bar b)^2}=1-2\delta \bar b+3\delta \bar b^2-4\delta \bar b^3+...$.
Then find the expected value of $\bar a/ \bar b^2$:
$E[\bar a/ \bar b^2]=\frac{\bar A}{\bar B^2}E[(1+\delta\bar a)(1-2\delta \bar b+3\delta \bar b^2-4\delta \bar b^3+...)]$
$E[\bar a/ \bar b^2]=\frac{\bar A}{\bar B^2}E[(1+\delta\bar a-2\delta \bar b-2\delta \bar b\delta\bar a+3\delta \bar b^2+3\delta \bar b^2\delta\bar a-4\delta \bar b^3-4\delta \bar b^3\delta\bar a+...)]$
We can approximate this by considering only second order effects (and noting that the expectation of the first three terms are 1, 0 and 0).
$E[\bar a/ \bar b^2]\approx\frac{\bar A}{\bar B^2}E[(1-2\delta \bar b\delta\bar a+3\delta \bar b^2)]=\frac{\bar A}{\bar B^2}\left(1-2\frac{cov(a,b)}{n\bar a \bar b}+3\frac{var(b)}{n\bar b^2}\right)$
where $cov(a,b)$ and $var(b)$ are unbiassed sample covariance and variance estimates.
So, we can make a corrected estimator 
$\hat E=\frac{\bar a}{\bar b^2}\left(1+\frac{2cov(a,b)}{n\bar a \bar b}-\frac{3var(b)}{n\bar b^2}\right)$
You might be able to make a better estimator by using more terms from the Taylor expansion, although it would require higher order cumulants which may be harder to estimate accurately. You might also be able to make a better estimator with the additional information about the distributions of $A$ and $B$.
