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