st? Suppose T01\") is sufficient for 3. Then, for any decision procedure 6{X} one can always find a randomized decision rule 5]"{X} = yfftXD. which depends only on THE} and does as well as ritX) in the sense of having the same risk function, i.e., mar?) : RtHTL for all 3 E 8. Hence. the decision can be made based directly on the sufficient statistics NIX) instead. .23? Normal distribution Suppose the sample X 2 (X1. . -. .33.} is from X ~ Hm, I). II The joint density ptmi = 171;; ex]:- 25;, r} + e 3;, e} for all 1: e a" and s e a; I The sample mean RX) = X is sufficient to the population mean (1'; I For {EX} = .15., set riftX) = 25;] LII-X.- + If? for {a],-~- .an) such that 2le a.- = f} and 2le a? = {n Ufa. Then. {o'TtX} IX = r} v Ntr.[n - 1}Ji'n].Tth1, Eloy-{Kl} = Emmy-{X} | 3]] = E[J?] = H = E[X]], 1 1 n + = 1 =VaI[X|]. H H 11-] Var[:'ir[X]I] = E[Vm-[:ST(X) | X]j+ye:[s[sytxt | in] = El |+VaI[J_E'] = H I As a consequence, air and :3 have the same MSE - risk. function. A. I! III Home-mans 5: Verify the equality between MSE of or and that of :5 in the above example. IIirfIJIJ [333s hated Tsdtcgy: Mics-ad F'etfxrrL-ILE