Take the tour ×  Suppose that one can construct an unbiased
estimator $X$ of the quantity $E$, is there a way of getting an
unbiased and positive estimator of $E^2$? Indeed, if $X_1$ and
$X_2$ are two independent copies of the estimator $X$, the
quantity $X_1 X_2$ is an unbiased estimator of $E^2$, but is not
positive in general.  Motivation: when interested in the
marginal distribution $\pi(\theta) = \int_{X} \pi(\theta, x) \,
dx$ of an intractable distribution $\pi(\theta,x)$, one can use
an MCMC algorithm if one can compute $\pi(\theta)$. This is
generally not the case. Nevertheless, it is now well-known that
this can also be done if one can only construct an unbiased and
positive estimator of $\pi(\theta)$, hence the question.
share|improve this question Yeah, that is always a problem with
unbiased estimators, they are not invariant and you have to find
'ad hoc' estimators. –  user10525 Apr 26 '12 at 23:19 In the
fully nonparametric case, the uniformly minimum variance
unbiased estimator is $\frac{2}{n(n