3.设T是g(θ)的UMVUE,g是g(θ)的无偏估计,证明,若 _(ar)(g)lt infty , 则 _(0)V(T,g)geqslant 0.-|||-4.设总体 sim N(mu ,(sigma )^2) 1,x2,···,xn为样本,证明 overline (x)=dfrac (1)(n)sum _(i=1)^n(x)_(i) ^2=dfrac (1)(n-1)sum _(i=1)^n(({x)_(i)-overline (x))}^2 分别为-|||-μ,σ^2的UMVUE.-|||-5.设总体p(x;θ)的费希尔信息量存在,若二阶导数 dfrac ({partial )^2}(partial {partial )^2}p(x;theta ) 对一切的 theta in bigcirc (1) 存在,证明费希尔-|||-信息量-|||-(theta )=-E(dfrac ({alpha )^2}({alpha )^2})pp(x;theta ))-|||-6.设总体密度函数为 (x;theta )=theta (x)^theta -1 lt xlt 1 theta gt 0, x1,x2,···,xn是样本.-|||-(1)求 (theta )=1/theta 的最大似然估计;-|||-(2)求g(θ)的有效估计.-|||-7.设总体密度函数为 (x;theta )=dfrac (2theta )({x)^3}(e)^-theta /(x^2) gt 0 theta gt 0, 求θ的费希尔信息量I(θ).-|||-8.设总体密度函数为 (x;theta )=theta (c)^theta (x)^-(theta +1) gt c gt 0 已知, theta gt 0, 求θ的费希尔信息量I(θ).-|||-9.设总体分布列为 (X=x)=(x-1)(theta )^2((1-theta ))^x-2 =2, 3,···, lt theta lt 1, 求θ的费希尔信息量I(θ).-|||-10.设x1,x2,···,xn是来自Ga(α,λ)的样本, gt 0 已知,试证明, overline (x)/a 是 (lambda )=1/lambda 的有效估计,-|||-从而也是UMVU JE.-|||-11.设x1,x2,···, _(m)in.d.sim N(a,(sigma )^2) y1,y2,···, _(n)i.dot (1)cdot d.sim N(a,2(sigma )^2), 求a和σ^2的UMVUE.-|||-12.设x1,x2,··· _(n)i.dot (1)cdot d.sim N(mu ,1), 求μ^2的UMVUE.证明此UMVUE达不到 C-R 不等式的下界,-|||-即它不是有效估计.