Let X_1, X_2,...,X_n be a random sample from a geometric distribution, with probability function given by

p_x(x)=Pr(X=x)=\frac{1}{a}(1-\frac{1}{a})^{x-1}, x=1,2,3,... where a>1

Now I can show that the maximum likelihood estimator of a is given by the sample mean, and have found its mean and variance. Which I think is given by


However, I am not sure how to show whether this estimator has the minimum variance for an unbiased estimator of a (is it the minimum variance unbiased estimator of a) or determine whether the maximum likelihood estimator of a is mean square consistent.