課程名稱︰高等統計推論
課程性質︰應數所數統組必修
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）︰2016/5/19
考試時限（分鐘）：16:30~17:20
試題 :
1. (15%) Let X ,...,X be a random sample from the uniform distribution
1 n
U(α-β,α+β) where α and β are unknown parameters. Find the uniformly
minimum variance unbiased estimator of α/β.
2. (10%) Let X ,...,X be a random sample from Poisson(λ) and λ have a
1 n
Gamma(α,β) distribution. Find the Bayes estimator of λ under the absolute
error loss function.
^ ^
3. (15%) Let δ be the Bayes estimator with the constant risk. Show that δ is
the minimax estimator.
4. (5%)(15%) State the necessary assumptions for the asymptotic normality of
maximum likelihood estimator and show this asymptotic property.
5. (15%) Let X ,...,X be a random sample from a population with probability
1 n
θ-1
density function f (x|θ)=θx , 0<x<1, 0<θ<∞. Derive the asymptotic
X
distribution of the maximum likelihood estimator of θ.
6. (15%) Let X ,...,X be a random sample from f(x-θ), where f(u) is symmetric
1 n
^
around zero. Let θ be the maximum likelihood estimator of θ and the
^ n
M-estimator θ be the minimizer of Σ ρ(X -θ), where ρ(u) is strictly
M i=1 i
convex with dρ(u)/du = φ(u). Compute the asymptotic relative efficiency of
^ ^
θ and θ.
M
7. (10%) Show that a uniformly most powerful level α test is an unbiased test.