Let $Y_1,Y_2,...,Y_n$ be i.d.d. random variables from a gamma distribution with shape parameter $\ \ \ \ \ \ \ \ \ \alpha_0>0$ and unknown scale parameter $\beta$. Find a $(1-\alpha)$ confidence interval for the parameter $\beta$. Note that the minimum variance unbiased estimator for $\beta$ is $\hat{\beta}=\frac{\bar{Y}}{\alpha_0}$ where $\bar{Y}=\frac{1}{n}\sum^n_{i=1}Y_i$.

I have no clue where to start here. Any hints?


Well, I bet you have to use the central limit theorem. The random variables $Y_1, Y_2, Y_3, ..., Y_n$ satisfy the condition of this theorem (they have the finite variance and same expectation $\mu$. Thus, their sum will have the following distribution (for any sufficiently large number $n$):

$$ P(\frac{\sum^n_{i=1}Y_n}{n})=N(0,\sigma^2) $$

Thus, $\bar{\beta}=\frac{1}{\alpha_0} \bar{Y}$ will have the following distribution: $P(\bar\beta)=N(0,\frac{\sigma^2}{\alpha_0})$. From the latter, you can get the confidence interval.

PS How to improve my answer if the variance is unknown, I don't have any precise ideas except that this may follow chi-squred distribution.


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