Let Y1, Y2,..., Yn denote a random sample from a Poisson-distributed population with
Chapter 16, Problem 16.11(choose chapter or problem)
Let Y1, Y2,..., Yn denote a random sample from a Poisson-distributed population with mean . In this case, U = Yi is a sufficient statistic for , and U has a Poisson distribution with mean n. Use the conjugate gamma (, ) prior for to do the following. a Show that the joint likelihood of U, is L(u, ) = nu u!() u+1 exp + n + 1 . b Show that the marginal mass function of U is m(u) = nu(u + ) u!() n + 1 u+ . c Show that the posterior density for | u is a gamma density with parameters = u + and = /(n + 1). d Show that the Bayes estimator for is B = Yi + n + 1 . e Show that the Bayes estimator in part (d) can be written as a weighted average of Y and the prior mean for . f Show that the Bayes estimator in part (d) is a biased but consistent estimator for .
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