Question.

Let be a random sample from a distribution with density function

where is a known integer, and is an unknown parameter. Using the asymptotic distribution of the maximum likelihood estimator, construct an interval estimator for with approximate confidence coefficient of 0.95.

Answer.

Asymptotic distribution of MLE: if is a maximum likelihood estimator, then converges in distribution to and is asymptotically pivotal for .

I found information for Gamma distribution random sample already here

http://www.mathhelpforum.com/math-he...tml#post617259

so I'll just use it here: . Since I don't know , I am going to use to approximate information.

Let

right?

Solving for , I get