## an unbiased estimator of the Hellinger distance?

Here is a question I posted on Stack Exchange a while ago:

In a setting where one observes X1,…,Xn distributed from a distribution with (unknown) density f, I wonder if there is an unbiased estimator (based on the Xi‘s) of the Hellinger distance to another distribution with known density f0, namely

$\mathfrak{H}(f,f_0)=\left\{1-\int\sqrt{f_0(x)/(x)}\text{d}x\right\}^{1/2}$
Now, Paulo has posted an answer that is rather interesting, if formally “off the point”. There exists a natural unbiased estimator of if not of H, based on the original sample and using the alternative representation
$\mathfrak{H}^2(f,f_0)=1-\mathbb{E}_f[\sqrt{f_0(X)/f(X)}]$

for the Hellinger distance. In addition, this estimator is guaranteed to enjoy a finite variance since

$\mathbb{E}_f[\sqrt{f_0(X)/f(X)}^2]=1\,.$

Considering this question again, I am now fairly convinced there cannot be an unbiased estimator of H, as it behaves like a standard deviation for which there usually is no unbiased estimator!