bias/unbiased estimator

**rlkmg** · Dec 2nd 2010, 07:05 AM

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

**Guy** · Dec 2nd 2010, 12:58 PM

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

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

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

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

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

**matheagle** · Dec 2nd 2010, 10:28 PM

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

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