# Checking whether a maximum likelihood estimator is biased

So I have a Poisson distribution:

$V \sim \operatorname{Po} \left({\rho v}\right)$

and I've calculated the maximum likelihood estimator $\widehat{\rho} = \dfrac{\overline{v}}{v}$ from independent samples $v_{1}...v_{n}$. How do I test whether $\widehat{\rho}$ is unbiased?

Thanks

• Verify that ${\rm E}[\hat{\rho}]=\rho$. – Stefan Hansen Feb 28 '14 at 11:23
• @StefanHansen thanks, how do I find $E[\widehat{\rho}]$ if I don't know the distribution for it? – Taimur Feb 28 '14 at 11:25
• The expectation is linear and you know the expectation of $v_1,\ldots,v_n$. – Stefan Hansen Feb 28 '14 at 11:26
• @StefanHansen ahhh yes of course, thank you – Taimur Feb 28 '14 at 11:26

If $E(\hat{p})=p$ then the estimator is unbiased.

In your case, $E(\hat{p})$ is simply the mean of the estimators you got from each of your samples

• This answer is slightly wrong - an estimator is unbiased if its mean is the true parameter (note this is for a fixed sample size). You do not take the sample size to infinity to check if an estimator is biased (if it becomes unbiased as the sample size goes to infinity, the estimator is said to be asymptotically unbiased). – Batman Mar 1 '14 at 5:32
• it's not the sample size, it's the number of samples ! – yakoudbz Mar 2 '14 at 22:01
• what??????????? – Batman Mar 3 '14 at 0:11
• sorry, I understand what I wrote didn't follow the definition of a unbiased estimator (I changed my answer accordingly), but in real life, if n is not a bit high, you can tell nothing about your estimator because you may just have took a bad or good sample... – yakoudbz Mar 3 '14 at 13:03