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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