※ 引述《il0306 (MrSix)》之銘言：
X1 , ... , Xn iid N(μ,σ^2)
T(X) = (X,S^2)
with S^2 = Σ(Xi-X)/(n-1)
T(X) is sufficient for (μ,σ^2) and complete
1) 如果 μ=0 ie N(0,σ^2)
T(X) 會sufficient for σ^2 ? 會 complete 嗎?
2) 如果 μ^2=σ^2 ie N(μ,μ^2)
T(X) 會sufficient for μ ? 會 complete 嗎?
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For Question2 :
iid
X1 ... Xn → N(θ,θ^2)
L(θ,x ) = (e^-1/2 / (2π)^-1/2 ) [(θ^2)^-1/2 ] exp{[(-x^2/2θ^2)+(x/θ)}
￣ ￣
so ( Σ(Xi)^2 , ΣXi ) is (minimal) sufficient statistic for θ
since T = (n-1)S^2 / θ^2 → Chi-Square(n-1)
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E((√n-1)S/θ) = c , where c is a constant
Noted that c can be derived from E(√T)
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so E((√n-1)S/c) = θ
╴ ___
Let g (Σ(Xi)^2 , ΣXi) = Xn- √n-1)S/c
Hence E(g(Σ(Xi)^2 , ΣXi)) = 0
but g is not zero function
╴
so T(X ) is not complete
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