I am triying to find an unbiased estimator for the variance of an exponential distribution.
Let we have a exponential distribution $f(x;\lambda) = \frac{1}{\lambda}e^{\frac{-x}{\lambda}} $.
I derived the MLE for the variance (which is also $\lambda^2$) as $\hat{\lambda^2} = ( \frac{\sum x_i}{n})^2 $
When I computed the bias $B(\hat{\lambda^2})$, I arrived to $\frac{\lambda^2}{n} $, which will make the estimator biased. So I must correct the estimator in order to make it unbiased, but I don't know how could I make the correction. I will appreciate any tips in how to proceed.
Thanks!
When you computed the bias, you must also have obtained the expectation of the estimator; i.e., $$\operatorname{E}\left[\widehat {\lambda^2}\right] = \frac{n+1}{n} \lambda^2.$$ So in order to correct for the bias, all you do is multiply the estimator by $\frac{n}{n+1}$; that is to say, $$\operatorname{E}\left[\frac{n}{n+1} \widehat {\lambda^2}\right] = \operatorname{E}\left[\frac{1}{n(n+1)} \Bigl( \sum_{i=1}^n X_i \Bigr)^2 \right] = \lambda^2.$$