d. It can be shown (you don't have to) that E (tlnX;) -I = -a/In - 1).

(See W. C. Guenther (1967), "A best statistic with variance not equal to the Cramer-Rao lower bound." American Mathematical Monthly, 74, pp. 993-994, or else you can derive the density of

('L7=llnX;)-1 and find its expectations-the density of 'L7=1 InX; is the mirror image (around the vertical axis at zero) of a Gamma density.) Is the MLE the MVUE for a? If not, is there a function of the MLE that is MVUE for a?

'e. Show that the MLE is asymptotically normal and asymptotically efficient, where n l / 2 (& -

a) -4 N(0,a2 ).