Let $X_1, \ldots, X_n$ be a random sample on an exponential distribution with mean $\theta$.
Obtain an unbiased estimator for $\theta$ based on $G$, where $G$ is the geometric mean of the observations.
Hint: answer may be expressed in terms of the gamma function.
Approach: I understand that I can obtain the MLE estimate for theta by differentiating the log-likelihood functions, and letting it = 0. This gives me $\hat{\theta} = \frac{1}{n} \sum_{i=1}^n X_i$.
I don't see how to bring $G$ or the gamma function into the answer
Any advice would be appreciated
self-study
tag and read its tag wiki. $\endgroup$ – Glen_b -Reinstate Monica Sep 21 '14 at 6:20