Determine if the MME of $X~Beta(1,\theta)$, a special case of beta distribution, is unbiased.

Question

Pdf of $X$ is $f(x) = \theta(1-x)^{\theta-1}$, $x\in[0,1]$.

I first evaluated the mme to get $\theta_{MME}=\frac{1}{\bar{X}}-1$.

Now I'm not sure how to proceed with determining if the expectation is equal to $\theta$ or not since I don't know anything about the distribution of $\bar{X}$ or $\bar{X}^{-1}$. I tried deriving the CDF of $\bar{X}$ but end up with something that looks very messy. Since this is a practice problem from a Uni test and the expected amount of time to solve it is about 20 min, I figured there must be something much simpler that I'm missing. Any ideas?

Edit: if the estimator was mle I would use the invariance property and the problem would be easy to solve. But I don't think mme shares this invariance property with mle.

Answer 1

Here is the solution in case anybody wants to know:

Since $f(x) = \frac{1}{x} - 1$ is (strictly) convex on $x>0$, we can apply Jensen Inequality to get that $$E[f(\bar{X})] > f(E[\bar{X}])$$

where LHS: $$E[f(\bar{X})] = E[\frac{1}{\bar{X}} -1] = E[\theta_{MME}]$$ and RHS: $$f(E[\bar{X}])=f(\frac{1}{1+\theta})=\frac{1}{\frac{1}{1+\theta}}-1=1+\theta-1=\theta$$

Thus $$E[\theta_{MME}]>\theta$$

So the estimator is biased.