Thread: bias/unbiased estimator

having trouble with question in bold, any help will be appreciated.. Thanks!

Just look for a function of $\displaystyle \hat{\theta}$ that is unbiased for $\displaystyle 1 / \theta$. A good start would be $\displaystyle 1 / \hat{\theta}$ and then fix it up so that it's unbiased. Running through this really quick, I got

$\displaystyle \hat{\phi} = \frac{2n - 1}{2n} \frac{1}{\hat{\theta}}$

which you can easily verify is unbiased and consistent. It'll help you out in doing this if you first verify the following: for $\displaystyle Y \sim G(\alpha, \beta)$, under the parametrization of the gamma given,

$\displaystyle \displaystyle
\mbox{E}[Y^r] = \frac{\Gamma(\alpha + r) \beta^r}{\Gamma(\alpha)}
$

for all r such that $\displaystyle \alpha + r > 0$.

for consistency (weak or strong) use the weak or strong law of large numbers for the sample mean.