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. Let Y(1) = min(Y1,...,Yn) be the smallest observation in the sample. Show that Y(1) has an exp(n?) distribution.

B. Show that the following two estimators are both unbiased estimators of ?:

?1 hat = nY(1) and ?2 hat = Y

where Y is the sample mean.

C. Derive the relative eciency of ?1 hat compared to ?2 hat. Which estimator do you prefer?

D. Determine whether ?1 hat and ?2 hat are consistent estimators of ?