next we find the Bayes estimator of a which is the posterior mode if the prior is proportional to 1/a2
so for this prior the estimator from the Bayesian method is essentially the same as the mle.
Say X1, .., Xn iid F with f(x|a)=axa-1, x>0, a>0 (or simply X~Beta(a,1))
First let's find the method of moments estimator and the maximum likelihood estimator of a:
next we find the Bayes estimator of a which is the posterior mode if the prior is proportional to 1/a2
so for this prior the estimator from the Bayesian method is essentially the same as the mle.
What properties do these estimators have?
Unbiasedness
is 
/(1-) unbiased for a? To find out we would need to first find the density of , but in this case that is not possible in this generality. In beta.ex(n=20) we run a simuation which seems to suggest that the estimator is biased for larger a.
How about the mle?
so the mle is almost unbiased.
Sufficiency:
so the mle is a sufficient statistic for a
Ancillary Statistic
so an/T is an ancillary statistic.
Consistency:
From the WLLN we know that →E[X1]=a/(a+1), so
Note: it is not true that if Xn→x in probability then g(Xn)→g(x) for any function g.
How about the mle? Again from the WLLN we have
and so the mle is a consistent estimator of a as well.
Relative Efficiency:
The relative efficiency of the two estimators is the ratio of their variances, unfortunately the variance of the method of moments estimator can not be calculated directly. We can, though, apply our approximation theorem:
Of course there is the question how good the approximation is. Running beta.ex(a,n) a couple of times for different values of a and n shows it to be quite good.
Here the MLE:
and again beta.ex(a,n) verifies that the approximation is pretty good. beta(2,a,n) veryfies some of these calculations.
Relative Efficiency:
beta.ex(3) draws the efficiency for different values of n and a, and we see that MLE is more efficient than the MM for larger values of n and a.
Rao-Cramer lower bound:
so neither the MM nor the MLE achieve the lower bound, although
so the MLE is asymptotically efficient. beta.ex(4) draws the graph of the variances and the Rao-Cramer lower bound for several values of n and a.
Robustness: in this case 0<x<1, so robustness is not an issue