则称U()为之一不偏估计式

1、不偏性 unbiasednessIf be a random sample from f(x;), 且 U() is an estimator of 當EU()=, 則稱U()為之一不偏估計式.Ex 1： be a random sample from f(x;)= , x>0 令U()= 檢驗結果U是否為之不偏估計式. Sol：EU()=E() = = = (而E(X)=) 檢驗結果U is a unbiased estimator for .Ex 2：If be a random from f(x;)=, x=0,1,2, Set g()= find a unbiased estim

2、ator of . Sol：step 1：探討之意義. step 2：找可能的不偏估計式. 原來之意義為Pr(X=0)= 令U()=1, if X=0 0, other EU()=1 =Pr(X)= U() is a unbiased for . 順道求Var(U)=E(U)-E(U) = E(U)- E(U) =-() =(1-)Rao-BlackwellIf be a random sample from f(x;), 當T() is a suff statistic for 且 U() is a unbiased estimator of 但U非為T之函數時Consider 則：(1)依

3、舊是一之不偏估計式 (2)Var()Var(U) (在U及T之基礎上，找到較為優越之估計式)在Ex 2中：f(x,x;)= =() =g(t;) where g(t;)=, h()= t=By F-N因式分解定理知， T= is a suff for T is also suff for =By R-B, 求進行處理：當T=t時, =Pr(U=1|T=t)=(*) T=poisson(n) (U=1 T=t)=(X=0 ) =( X=0 ) (*)= =()=(1-)結論：由R-B所找到的 =(1-) =(1-) Var()Var(U)proof (R-B)：If =E(U|T) check

4、E()= (?) E()=E(E(U|T)=E() = = = = = E(E(U|T)= = = = = =check Var()Var(U) (?) ex：年級 1 2 3 4 50 70 50 60 60 80 70 7070 90 90 80 平均 60 80 70 70 總平均70 i=1,2,3 j=1,2,3,4Var(U)=Var()=3(60-70)+3(80-70)+3(70-70)+3(70-70)/12比較 = = =3(60-70)+3(80-70)+3(70-70)+3(70-70)Sol：Var(U)=Var(E(U|T)+E(Var(U|T) Var(U)=Var()+E(Var(U|T) E(Var(U|T)0 Var(U)Var() 令=E(U|T) Var(U)=E(U-E(U) =E(U-+-E(U) =E(E(U-+-E(U)|T) =EE(U-)|T+E(E(-E(U)|T) Var(E(U|T)=E(-E() =E(-E(U) =E(E(-E(U)|T) 另外：E(Var(U|T) Var(U|T)=E(U-E(U|T)|T) E(Var(U|T)=E(E(U-)|T)H.w. If be a random from f(x;)=, x=0,1,2, (1)找一個un