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

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

$\displaystyle E(X_{bar})=a $

$\displaystyle Var(X_{bar})=\frac{a(a-1)}{n} $

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.