6. A large commercial bank intends to analyze the accuracy with which their bank tellers process cash transactions. In particular, it desires an estimate of the expected proportion of daily cash transactions that the bank tellers process correctly. It plans to analyze 200 past observations on the daily proportion of correct cash transactions by the tellers, and specify the statistical model underlying the daily observations as:

fðz; aÞ ¼ aza1Ið Þ 0;1 ðzÞ; a 2 ð0;1Þ

(i.e., f(z;a) is a beta density function with b ¼ 1).

(a) Define the maximum likelihood estimator of a.

(b) Show that the MLE is a function of the complete sufficient statistic for this problem.

(c) Is the MLE of a a consistent estimator?

(d) It can be shown (you don’t have to) that E Xn i¼1 lnð Þ Xi

1

¼ a=ð Þ n  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 Pn i¼1 lnðXiÞ  1 and find its expectations—the density of Pn i¼1 lnðXiÞ 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 n1=2 ð^a  aÞ!d Nð0; a2Þ.

(f) Define the MLE of q

(a) ¼ a/(a + 1), which is the expected proportion of correct cash transactions.