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

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

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!

3 Responses to “an unbiased estimator of the Hellinger distance?”

  1. […] 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. […]

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

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