課程名稱︰高等統計推論二 課程性質︰數學系選修 課程教師︰江金倉 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/4/21 考試時限（分鐘）：15:30~17:30 試題 : iid 2 1. (15%) Let X = θ X +ε, i=1,...,n, with X =ε and ε,...,ε ~ N(0,σ ). i 0 i-1 i 1 1 1 n 0 2 Find the maximum likelihood estimator of (θ,σ ). 0 0 n Σ X X i=2 i i-1 1 n 2 (——————, — Σ (X - θX ) ) n-1 2 n i=1 i 0 i-1 Σ X i=1 i-1 2. (15%) Conditioning on Z = z, T and C are assumed to be independent with T T following an exponential distribution with rate exp(β z) and C being 0 non-informative about β . Based on a random sample of the form 0 n {X ,δ,Z } , where X = min{T ,C } and δ=I(X =T ), find the maximum i i i i=1 i i i i i i likelihood estimator of β . 0 Hint: You just need to write down how to calculate. 3. (10%) Let X ,...,X be a random sample from a population with a probability 1 n θ-1 0 density function f(x|θ) = θx I(0<x<1), where θ>0. Show that the 0 0 0 variance of the UMVUE of θ cannot attain the Cramer-Rao lower bound. 0 4. (15%) Let X ,...,X be a random sample from a Uniform(θ,θ) with θ<θ. 1 n 1 2 1 2 Find the uniformly minimum variance unbiased estimator of the range R = θ-θ. 2 1 5. (15%) Let T and T be sufficient statistic and minimum sufficient statistic, 1 2 respectively. How are E[W |T ] and E[W |T ] related? n 1 n 2 6. Let X ,...,X be a random sample from N(θ,aθ), where a is a known positive 1 n 0 0 constant and θ> 0. 0 (6a) (7%) Find a minimal sufficient statistic for θ. (6b) (6%)(7%) For any constant 0≦c≦1, find d such that _ 2 ^ E[cX +(1-c)dS |θ]=θ and find the minimizer, say c, of n n _ 2 Var(cX +(1-c)dS |θ). n n 7. (5%)(10%) State and show the Lehmann-Scheffe theorem. ^ 8. (10%) Let θ be a sufficient statistic and the unique maximum likelihood ^ estimator for θ. Show that θ is the minimum sufficient statistic for θ. 0 0 f(X|θ)=g(T(X),θ)h(x). So maxarg f(X|θ)=maxarg g(T(X),θ). ~ ~ ~ θ ~ θ ~ ^ ^ Thus, θ is a function of T(X). So, θ is minimum. ~ --

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