3.23 If X1,...,Xn are iid P(), consider estimation of eb, where b is known. (a) Show that...

3.23 If X1,...,Xn are iid P(λ), consider estimation of e−bλ, where b is known.

(a) Show that δ∗ = (1 − b/n)

t is the UMVU estimator of e−bλ.

(b) For b>n, describe the behavior of δ∗, and suggest why it might not be a reasonable estimator.

(The probability e−bλ, for b>n, is that of an “unobservable” event, in that it can be interpreted as the probability of no occurrence in a time interval of length

b. A number of such situations are described and analyzed in Lehmann (1983), where it is suggested that, in these problems, no reasonable estimator may exist.)