Let $X$ and $Y$ be two independent random variables. The product of their variances is:
$\mu_{2,0}\mu_{0,2} = E[(X-E[X])^2]E[(Y-E[Y])^2]$
We know an unbiased estimator of variance is:
$\hat{\mu}_{2,0} = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2$
$\hat{\mu}_{0,2} = \frac{1}{n-1}\sum_{i=1}^n (y_i - \bar{y})^2$
So, is $\hat{\mu}_{2,0} \hat{\mu}_{0,2}$ an unbiased estimator of $\mu_{2,0}\mu_{0,2}$?
Numerical experiments tell me that it is. But I also know that there is something called multivariate Polyache (not yet implemented in mathStatica). I just want to make sure if this is a special case or not.
Yes, since the one estimator only depends on samples of $X$ and the other of $Y,$ the two estimators are independent and the expectation of their products is the product of their expectations. So unbiasedness of the product follows from the two factors being unbiased.