PRINCIPLES OF STATISTICS - EXAMPLES 3/4 Part II, Michaelmas 2016, Quentin Berthet (email: q.berthet@statslab.cam.ac.uk) Throughout, for observations X arising from a parametric model {f (., 0) : 0 € 0}, O C R, the quadratic risk of a decision rule &(X) is defined to be R(8, 0) = Ee(8(X) - 0)2. 1. Consider X|0 ~ Poisson (0), 0 € O = [0, 00), and suppose the prior for 0 is a Gamma distribution with parameters a, A. Show that the posterior distribution 0|X is also a Gamma distribution and find its parameters. 2. For n E N fixed, suppose X is binomially Bin(n, 0)-distributed where 0 € O = [0, 1]. a) Consider a prior for 0 from a Beta(a, b), a, b > 0, distribution. Show that the posterior distribution is Beta(a + X, b + n - X) and compute the posterior mean 0m(X) = E(0|X). b) Show that the maximum likelihood estimator for 0 is not identical to the posterior mean with 'ignorant' uniform prior 0 ~ U[0, 1]. c) Assuming that X is sampled from a fixed Bin(n, 00), 00 € (0, 1), distribution, derive the asymptotic distribution of Vn(On(X) - 00) as n > co. 3. Let X1, ... , Xn be i.i.d. copies of a random variable X and consider the Bayesian model X|0 ~ N(0, 1) with prior T as 0 ~ N(u, v2). For 0 < < < 1, consider the credible interval Cn = {0 € IR : |0 - E™ (0\X1, ., Xn)| < Rn} where Rn is chosen such that T(Cn|X1, ... , Xn) = 1 - a. Now assume X ~ N(00, 1) for some fixed 00 € R, and show that, as n -> 00, Po (00 € Cn) -> 1 - a. 4. In a general decision problem, show that a) a decision rule & that has constant risk and is admissible is also minimax; b) any unique Bayes rule is admissible. 5. Consider an observation X from a parametric model {f (., 0) : 0 € O} with prior T on O C R and a general risk function R(8, 0) = EL(8(X), 0). Assume that there exists some decision rule do such that R(80, 0) < > for all 0 € O. a) For the loss function L(a, 0) = |a - 0| show that the Bayes rule associated to T equals any median of the posterior distribution T (.|X). b) For weight function w : O -> [0, 00) and loss function L(a, 0) = w(0)[a - 0]2 show that the Bayes rule &7 associated to T is unique and equals ÔT (X) = ET [w(0)| X] ' ET [w(0)0| X] assuming that the expectations in the last ratio exist, are finite, and that E™ [w(0)|X] > 0. 6. a) Considering X1, ... , Xn i.i.d. from a N(0, 1)-model with 0 E O = R, show that the maximum likelihood estimator is not a Bayes rule for estimating 0 in quadratic risk