Unbiasedness of product/quotient of two unbiased estimators An answer to this question might just be "it depends", however I am wondering:
Given unbiased estimators $\hat{\mu}_X$ and $\hat{\mu}_Y$ for the means $\mu_X$ of $X$ and $\mu_Y$ of $Y$ respectively.  Under what conditions are the following true: $$E[\hat{\mu}_X\hat{\mu}_Y]=\mu_{XY}$$ or $$E[\frac{\hat{\mu}_X}{\hat{\mu}_Y}]=\mu_{\frac{X}{Y}}$$
Is the first one always true when $X \perp\ Y$?  I don't have any guesses yet about the second part.  Thank you.
 A: You need to impose more restrictive conditions, even for the first result. Suppose $X$ and $Y$ are orthogonal (even more: independent) with zero mean, so $E(XY)=0$. Take the unbiased estimators $\hat\mu_X=X_1 + Y_1$, $\hat\mu_Y=Y_1$ Then $E[\hat{\mu}_X \hat{\mu}_Y]=\sigma_Y^2 \ne 0$. A sufficient requirement of the property to hold would be that $X$ $Y$ are independent and $\hat{\mu}_X$ (  $\hat{\mu}_Y$)is a function of samples of $X$ ($Y$) only; which would imply that the estimators themselves are independent. The independence condition could be relaxed to orthogonality in some cases (eg if the estimators are linear).
The same conditions would even not be enough for the second. One would like to write
$$E[\frac{\hat{\mu}_X}{\hat{\mu}_Y}]=E[\hat{\mu}_X]E[\frac{1}{\hat{\mu}_Y}]= \mu_X \frac{1}{\mu_Y} = E[\frac{X}{Y}]$$
but all those equalities are false in general.
Assuming the conditions above (both $X$ $Y$ and the estimators are independent), then we'd need in adition that this holds: $E[1/Y] = 1/E[Y]$ (and the same for the estimator). This is false in general - it can be approximately true if $Y$ has small variance.
