Consider a sample of n > 2 independent and identically distributed random variables Y1,...,Yn from an exp(?) distribution with probability density function f(y;?) = ?exp(??y) y > 0 ? > 0. Use the notation ? = 1 /?.

Take into consideration, the following 2 facts:

(1) Let Sn =Sum n, i=1 Yi be the sum of n independent exp(?) random variables.

Then Sn has a Gamma distribution, Gamma(n,?), which has probability density function

f(s;n,?) = [?nsn?1 exp(??s)] / (n?1)!

s > 0 ? > 0 n ? 1.

(2) If S ? Gamma(n,?) then

E(1 /S)= ? (n?1)

Var(1 /S)= ?2 / [(n?1)²/ (n?2)

(for n > 2).

Question:

A. Write down the likelihood function and the log-likelihood function, as functions of ?.

B. Find a sucient statistic for ?.

C. Derive the maximum likelihood estimator (MLE) of ?. Comment on your result in relation to part (6).

D. Find the mean and variance of the MLE of ? and hence show that it is biased but consistent.

E. Obtain the expected information and hence write down an approximate large sample 95% condence interval for ?.

F. Based on the MLE, construct an unbiased estimator of ?.

G. What is the minimum variance bound for unbiased estimators of ??

H. Does the estimator you constructed in part (10) attain the minimum variance bound?

I. What is the MLE of ??