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）︰2016/6/16
考试时限（分钟）：16:30~17:20
试题 :
1. (15%) Consider the hypotheses H :θ∈Θ versus H :θ∈Θ. Let W(X ,...,X )
0 0 A 1 1 n
be a test statistic such that large values of W(X ,...,X ) give evidence that
1 n
H is ture. Define p(x ,...,x )=sup P(W(X ,...,X )≧W(x ,...,x )|θ). Show
A 1 n θ∈Θ 1 n 1 n
0
that p(X ,...,X ) is a valid p-value.
1 n
2. (20%) Let X ,...,X be a random sample from a p.d.f f(x|θ) with θ∈Θ and
1 n
dim(Θ)=k. Consider the null hypothesis H :θ∈Θ versus the alternative
0 0
hypothesis H :θ∈Θ-Θ, where Θ={θ:θ=g(η)}⊂Θ with η is a (k-r) ×1
A 0 0
unknown parameter vector and g(‧) is a continuously differential function from
k-r k d 2
R to R . Show that -2ln(λ )→χ under H and the regularity conditions,
n r 0
where λ is the likelihood ratio test statistic.
n
2 2
3. (15%) Let X ,...,X be a random sample from N(μ,σ) with σ being unknown.
1 n
Derive the power function of the size α likelihood ratio test for the null
hypothesis H :μ≦μ versus the alternative hypothesis H :μ>μ with the power
0 0 A 0
function being expressed via Φ (‧), which is the t distribution with ν
T,ν
degrees of freedom.
4. (10%)(10%) Let p (X ,...,X ) denote a valid p-value for H :θ∈Θ versus
j 1 n 0j j
H :θ not in Θ, j=1,...,k. Define a valid p-value p(X ,...,X ) for the
Aj j 1 n
k k c
hypotheses H :θ∈∪ Θ versus H :θ∈∩ Θ and find a level α test based on
0 j=1 j A j=1 j
p(X ,...,X ).
1 n
5. (10%) Let X ,...,X be a random sample from Uniform(θ,θ+1). Find the
1 n
uniformly most powerful size α test of the hypotheses H :θ=0 versus H :θ>0.
0 A
6. (10%)(10%) Let X ,...,X be a random sample from Bernoulli(p ) and Y ,...,Y
1 n 1 1 m
be another independent random sample from Bernoulli(p ). A hypothesis test of
2
interest is H :p = p versus H :p ≠p . Find a Wald test statistic and a score
0 1 2 A 1 2
test statistic.