Trading stamps. A trading stamp company wished to estimate the proportion p,) of stamps that will eventually be redeemed. Specially marked stamps were simultaneously released and their times to redemption were observed until a time yo after release. Suppose that the cumulative distribution of time to redemption of redeemed stamps is F(y). The cumulative distribution of time to redemption for all stamps is p,F( y); it is degenerate at infinity, since a proportion 1- po of the stamps are not redeemed. Assume (wrongly) that stamps are statistically independent and that F( y) is an exponential distribution with mean "failure" rate λ
(a) Give the expression for the probability a stamp is not redeemed
(b) Give the likelihood for r observed stamps (redemption time y,) in a sample of n stamps where n - r times are singly censored at time yo.
(c) Assume that yo is much greater than θ and simplify the like lihood so that it lacks yo. Give the log likelihood.
d) Calculate the likelihood equations for p and λ.
(e) Calculate the local estimate of asymptotic covariance matrix of the ML estimates
and λ. Leave in terms of and λ.