Statistics Discussion: Bayes Rules, Hypothesis Tests, Confidence Sets, and Consistency, Study notes of Mathematical Statistics

Download Statistics Discussion: Bayes Rules, Hypothesis Tests, Confidence Sets, and Consistency and more Study notes Mathematical Statistics in PDF only on Docsity! TA: Yuan Jiang Email: jiangy@stat.wisc.edu STAT 709: Discussion #17 November 8, 2007 1 Bayes Rules and Minimax Rules Example 1. Let X be a sample having a probability density fj(x) with respect to a σ-finite measure ν, where j is unknown and j ∈ {1, ..., J} with a known integer J ≥ 2. Consider a decision problem in which the action space is {1, ..., J} and the loss function is L(j, a) = { 0 if a = j 1 if a 6= j. (a) Obtain the risk of a decision rule (which may be randomized). (b) Let Π be a prior probability measure on {1, ..., J} with Π({j}) = πj , j = 1, . . . , J . Obtain the Bayes risk of a decision rule. (c) Obtain a Bayes rule under the prior Π in (b). (d) Assume that J = 2, π1 = π2 = 0.5, and fj(x) = φ(x − µj), where φ(x) is the Lebesgue density of the standard normal distribution and µj, j = 1, 2, are known constants. Obtain the Bayes rule in (c). (e) Obtain a minimax rule when J = 2. 2 Hypothesis Tests Example 2. Let X1, . . . , Xn be i.i.d. random variables having the exponen- tial distribution E(0, θ), θ ∈ (0,∞). Consider the hypothesis H0 : θ ≤ θ0 v.s. H1 : θ > θ0 where θ0 > 0 is a fixed constant. For the testing rule Tc(X) = 1(c,∞)(X̄): (a) Compute its size. (b) Find a cα such that Tcα has size α, where α is a given level of signifi- cance. (c) Find the p-value for Tcα. Office: 1275A MSC 1 Phone: 262-1577