XianJiaotongUniv.Prof.Cai Yuan-Li[证明]根据最小方差估计的定义可知B(x) = E[x -x(z)]T[x - x(z)]= J +[x -(z)]T[x -(2)]f(x,z)dxdz+0+αfz(z[x - x(z)]T[x - x(z)]f(x|z)dx d2Ofz(z)Bz(x)dz11
Xi’an Jiaotong Univ. Prof. Cai Yuan-Li 11 [证明] 根据最小方差估计的定义可知 𝐵(𝑥̃) = 𝐸[𝑥 − 𝑥̂(𝑧)] 𝑇 [𝑥 − 𝑥̂(𝑧)] = ∬ +∞ −∞ [𝑥 − 𝑥̂(𝑧)] 𝑇 [𝑥 − 𝑥̂(𝑧)]𝑓(𝑥, 𝑧)𝑑𝑥𝑑𝑧 = ∫ 𝑓𝑧 (𝑧) {∫ [𝑥 − 𝑥̂(𝑧)] 𝑇 [𝑥 − 𝑥̂(𝑧)]𝑓(𝑥|𝑧)𝑑𝑥 +∞ −∞ } 𝑑𝑧 +∞ −∞ = ∫ 𝑓𝑧(𝑧)𝐵𝑧(𝑥̃)𝑑𝑧 +∞ −∞
XianJiaotongUniv.Prof. Cai Yuan-Li不难发现B(x) = min 一 Bz(x) = min由极值必要条件+00aBz(x)-2(x -x)f(x|z)dx = 0ax00+80= xMV(z) =xf(x|z)dx = E[x|z]X12
Xi’an Jiaotong Univ. Prof. Cai Yuan-Li 12 不难发现 𝐵(𝑥̃) ⇒ min ⇔ 𝐵𝑍(𝑥̃) ⇒ min 由极值必要条件 𝜕𝐵𝑍(𝑥̃) 𝜕𝑥̂ = −2 ∫ (𝑥 − 𝑥̂)𝑓(𝑥|𝑧)𝑑𝑥 = 0 +∞ −∞ ⇒ 𝑥̂𝑀𝑉(𝑧) = ∫ 𝑥𝑓(𝑥|𝑧)𝑑𝑥 +∞ −∞ = 𝐸[𝑥|𝑧]
XianJiaotongUniv.Prof.Cai Yuan-Li验证充分条件a2Bz()2 = 21 > 0 (Positive)证毕！a28[推论3-1]最小方差估计的均值及估计误差协方差分别为(1)ExMv(z) = Ex(ExmV = 0)(2) Pxv = Jt Px(zf(z)dz = E,PxIz[证明]根据均值的定义13
Xi’an Jiaotong Univ. Prof. Cai Yuan-Li 13 验证充分条件 𝜕 2𝐵𝑍(𝑥̃) 𝜕2𝑥̂ = 2𝐼 > 0 (Positive) 证毕！ [推论 3-1] 最小方差估计的均值及估计误差协方差分别为 (1)𝐸𝑥̂𝑀𝑉(𝑧) = 𝐸𝑥 (𝐸𝑥̃𝑀𝑉 = 0) (2)𝑃𝑥̃𝑀𝑉 = ∫ 𝑃𝑥|𝑧𝑓𝑧 (𝑧)𝑑𝑧 +∞ −∞ = 𝐸𝑧𝑃𝑥|𝑧 [证明] 根据均值的定义
XianJiaotongUniv.Prof.Cai Yuan-Li0Ex|zx/zf,(z)dz+00xfx/z(x)fz(z)dxdzfxfxz(x,z)dxdz= x因此ExMv(z）=Ex，即ExmV=0.再由协方差的定义Pamv = ExXmvxNV14
Xi’an Jiaotong Univ. Prof. Cai Yuan-Li 14 𝐸𝑥|𝑧 = ∫ 𝑥|𝑧𝑓𝑧 (𝑧)𝑑𝑧 +∞ −∞ = ∬ 𝑥𝑓𝑥|𝑧 (𝑥)𝑓𝑧 (𝑧)𝑑𝑥𝑑𝑧 +∞ −∞ = ∬ 𝑥𝑓𝑥𝑧(𝑥, 𝑧)𝑑𝑥𝑑𝑧 +∞ −∞ = 𝑥̅ 因此𝐸𝑥̂𝑀𝑉(𝑧) = 𝐸𝑥，即𝐸𝑥̃𝑀𝑉 = 0. 再由协方差的定义 𝑃𝑥̃𝑀𝑉 = 𝐸𝑥̃𝑀𝑉𝑥̃𝑀𝑉 𝑇
Xian Jiaotong Univ.Prof.Cai Yuan-Li= E((x - x|z)(x - x|z)+80fz(z)dz(x - xz)(x - x|z)T fxlz(x)dxo00= EzPxz证毕！[推论3-2]若x与z相互独立，则Xmy = E[x|z] = E x = x.15
Xi’an Jiaotong Univ. Prof. Cai Yuan-Li 15 = 𝐸{(𝑥 − 𝑥|𝑧)(𝑥 − 𝑥|𝑧) 𝑇 } = ∫ 𝑓𝑧(𝑧)𝑑𝑧 +∞ −∞ ∫ (𝑥 − 𝑥|𝑧)(𝑥 − 𝑥|𝑧) 𝑇𝑓𝑥|𝑧 (𝑥)𝑑𝑥 +∞ −∞ = 𝐸𝑧𝑃𝑥|𝑧 证毕! [推论 3-2] 若𝑥与𝑧相互独立，则 𝑥̂𝑀𝑉 = 𝐸[ 𝑥|𝑧] = 𝐸 𝑥 = 𝑥̄