## an unbiased estimator of the Hellinger distance?

**H**ere is a question I posted on Stack Exchange a while ago:

In a setting where one observes

Xdistributed from a distribution with (unknown) density_{1},…,X_{n}f, I wonder if there is an unbiased estimator (based on theX) of the Hellinger distance to another distribution with known density_{i}‘sf, namely_{0}

**N**ow, Paulo has posted an answer that is rather interesting, if formally “off the point”. There exists a natural unbiased estimator of

*H²*if not of

*H*, based on the original sample and using the alternative representation

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!

October 22, 2012 at 2:44 pm

[…] Acabei de descobrir que não existe estimador não-viesado para o desvio-padrão. Sempre aprendi que havia um estimador não-viesado para a variância. Que os livros (e os professores) silenciassem sobre um estimador não-viesado para o desvio-padrão nunca me chamou a atenção. Afinal, parecia natural que haveria um estimador não-viesado para o desvio padrão: a raiz quadrada da variância amostral. Porém, isso não funciona. Felizmente sou Bayesiano e não me preocupo com o viés. […]

October 22, 2012 at 4:05 pm

Thanks. This is indeed of my favourite arguments against unbiasedness as a relevant criterion. There was another similar question on Stack Exchange about unbiased Bayes estimators (they do no exist!).

October 22, 2012 at 3:00 am

You intuition is probably right. We should have a proof of that.