Question below

(Bayesian estimation) Let X1, ..., X, be an i.i.d. sample drawn from the Burr distribution, with the cumulative distribution function F(x 0) =1- (1 + 12 )8' 1 20, (1) parameterized by the parameter d > 0. Assume that o has prior gamma distribution I(a, A), with the prior density T. (0) T(a) for some given a > 0 and A > 0. a) Prove that the posterior density also belongs to the family of gamma distributions and determine its parameters. b) Determine the Bayesian estimator g* for the function g(0) = 1/0, with respect to the Bayesian mean squared error Ex.e(9 - 9(0))2. (Sufficient statistics) Assume again that the data ~], ...,, comes from an independent sample X1, ...; Xn having the Burr distribution, i.e., the marginal cdf of X, is given by (1). Prove that the MLE e is a sufficient statistic for the parameter 0