课程名称︰高等统计推论二
课程性质︰应数所数统组必修
课程教师︰江金仓
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/4/11
考试时限（分钟）：11:20~12:10
试题 :
2
1. (10%) Let X ,...,X be a random sample from N(μ,σ ). Find the maximum
1 n
likelihood estimator of Φ((x-μ)/σ), where Φ(‧) is the cumulative
distribution function of the standard normal distribution and x is a given
value.
2. (10%) Suppose that Y ~ χ , i=1,...,k, are mutually independent. Try to find
i υ
i
k
the Satterthwaite approximation for Σ a Y , where a 's are known constant.
i=1 i i i
3. (15%) Let X ,...,X be a random sample from a p.d.f f(x|θ) with θ∈Θ and
1 n 0 0
_
dim(Φ)=1, θ be an unbiased estimator of θ which attains the Cramer-Rao lower
n
^ ^ 2
bound. Suppose that θ is a solution of ∂l(θ|X ,...,X ) = 0 and -∂l(θ|X ,..
n θ n 1 n θ 1
.,X ) > 0 for all θ, where l(θ|X ,...,X ) is the log-likelihood function.
n 1 n
^ _
Show that θ=θ.
n n
4. (15%) Let (Z ,δ),...,(Z ,δ) be a random sample with Z = min{X ,Y }, and
1 1 n n i i i
δ= I(Z = X ), i=1,...,n. Moreover, let X and Y be independent exponential
i i i
random variables with rates μ and λ. Find the maximum likelihood estimators
of λ and μ.
5. (20%) Show the monotonicity of expectation-maximization (EM) sequence.
6. (15%)(15%) Show that the uniformly minimum variance unbiased estimator
(UMVUE) is unique and uncorrelated with zero unbiased estimators.