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The positive random variables $X_{1}, X_{2},...X_{n}$ are independent observations having the Gamma distribution $Ga(3,\frac{1}{\eta})$, with density function:

$\frac{x^{2}}{2\eta^{3}}e^{\frac{-x}{\eta}}$ $(x>0)$

Which depends on the unknown parameter $\eta \in (0,\infty)$.

(i) Find the maximum likelihood estimator of $\eta$.

My solution:

$\hat \eta = \frac{\bar X}{3}$

(ii) Show that the maximum likelihood estimator of $\eta$ is unbiased and find its variance.

Please may someone explain (or give me a hint) how to prove the estimator is unbiased?

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An estimator is unbiased if its expectation equals the parameter; in this case, if $$\operatorname{E}[\hat \eta] = \operatorname{E}[\bar X/3] = \operatorname{E}[\bar X]/3 = \eta,$$ then $\hat \eta$ is unbiased for $\eta$.

Do you know how to determine $\operatorname{E}[X]$, the expectation of a single observation? If so, then use the linearity of expectation to compute $$\operatorname{E}[\bar X] = \operatorname{E}[(X_1 + X_2 + \cdots + X_n)/n].$$

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  • $\begingroup$ Thank you for your help. Please may you demonstrate how to use this method to find the variance? $\endgroup$
    – Connor Simmons
    Apr 17, 2022 at 20:34
  • $\begingroup$ @ConnorSimmons When $X_1, X_2, \ldots, X_n$ are IID, recall the formula $$\operatorname{Var}[X_1 + X_2 + \cdots + X_n] = \operatorname{Var}[X_1] + \operatorname{Var}[X_2] + \cdots + \operatorname{Var}[X_n].$$ $\endgroup$
    – heropup
    Apr 17, 2022 at 21:58
  • $\begingroup$ Again, I understand about the theory behind how to find the variance. It's just that I don't know how to do it in practice for this particular example. Any help would be greatly appreciated! $\endgroup$
    – Connor Simmons
    Apr 18, 2022 at 14:35

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