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,…,X_n distributed from a distribution with (unknown) density f, I wonder if there is an unbiased estimator (based on the X_i‘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 H² 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!

This entry was posted on October 22, 2012 at 12:12 am and is filed under Statistics with tags Hellinger loss, StackExchange, standard deviation, unbiasedness. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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

Estimador não-viesado para o desvio-padrão | Blog Pra falar de coisas Says:
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. [...]

Reply
- xi'an Says:
  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!).
  
  Reply
Paulo. Says:
October 22, 2012 at 3:00 am

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

Reply

Xi'an's Og