# 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