Unbiased Estimator for Gamma Distribution Parameter

[Legendre]

Homework Statement

I need to show that c / (sample mean X) is unbiased for b, for some c.

Where {X} are iid Gamma (a,b).

Homework Equations

N.A.

The Attempt at a Solution

I know how to show that (sample mean X) / a is unbiased for 1/b :

1) E(sample mean X / a) = (1/a)(1/n)(sum of E(X)) by iid
2) Then since E(X) = a/b, sum of E(X) = an/b. And we get the result.

But I have no idea how to evaluate E(1/ (sample mean X))! Where do I start? Thanks!

 

[mighty2000]

Hello there! To show that c / (sample mean X) is unbiased for b, you can use the same approach as before. First, let's remind ourselves of the definition of unbiasedness: a statistic is unbiased for a parameter if the expected value of the statistic is equal to the true value of the parameter.

So, let's start with the expected value of c / (sample mean X):

E(c / (sample mean X)) = c * E(1 / (sample mean X))

Now, we can use the fact that the sample mean is a consistent estimator for the true mean of the distribution. This means that as the sample size increases, the sample mean will converge to the true mean. In other words, as n approaches infinity, sample mean X approaches the true mean of the distribution, which we know is a/b (from the given information).

So, as n approaches infinity, E(1 / (sample mean X)) approaches E(1 / (a/b)), which is equal to b/a (using the same logic as before).

Therefore, we can say that as n approaches infinity, E(c / (sample mean X)) approaches c * (b/a) = (c/a) * b.

Since we know that b is the true value of the parameter, we can say that c / (sample mean X) is unbiased for b.

I hope this helps! Let me know if you have any other questions.