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）︰2016/6/23
考試時限（分鐘）：15:30~17:20
試題 :
1. (10%) Let X ,...,X be a random sample from a population with p.d.f f(x) and
1 n
c.d.f F(x), which is differentiable. Derive the asymptotic distribution of
√n(M -μ), where M and μ are the sample median and population median,
n n
respectively.
2. (10%) Let X ,...,X be a random sample from Bernoulli(p ) with n≧4. Find
1 n 0
4
the uniformly minimum variance unbiased estimator of p .
0
3. (5%)(10%) Let F (t) and F (c) denote separately the cumulative distribution
T C
functions of the non-negative continuous random variables T and C. Suppose that
T and C are independent and C is noninformative. Based on a random sample of
n
the form {(X ,δ)} , where X = min{T ,C } and δ= I(X = T ), write down the
i i i=1 i i i i i i
corresponding likelihood function and derive the maximum likelihood estimator
of F (t).
T
4. (15%) Let X ,...,X be a random sample from Uniform(θ,θ+1). Consider the
1 n
hypotheses H :θ=0 versus H :θ>0. Calculate the power of the unformly most
0 A
powerful size α test at any θ > 0.
1
2 2
5. (15%) Let X ,...,X be a random sample from N(μ,σ ), where μ and σ are
1 n
unknown. Find the constants a and b so that the (1-α) confidence interval of
2 2 2 2
the form {σ :(n-1)S /b≦σ ≦(n-1)S /a} has the minimum length.
n n
6. Let X ,...,X be a random sample from a population with the probability
1 n
density function f(x|λ)=λexp(-λx)1 (x), where λ is a positive
(0,∞)
parameter.
(6a) (10%) Find a uniformly most powerful size α, 0<α<1 test for the null
hypothesis H :λ=λ versus the alternative hypothesis H :λ>λ , where λ is
0 0 A 0 0
a known constant.
(6b) (10%) Find a uniformly most accurate (1-α) confidence interval for λ.
2
7. (5%)(10%) Let X ,...,X be a random sample from N(θ,σ ). Find an unbiased
1 n
size α test for the hypotheses H :θ≦θ≦θ versus H :θ<θ or θ>θ. Show
0 1 2 A 1 2
that the given test is unbiased.
8. (15%) Let X ,...,X be a random sample from a Poisson(λ) and λ have a
1 n
Gamma(α,β) prior distribution. Find a Bayes test of H :λ≦λ versus
0 0
H :λ>λ.
A 0
9. Let X ,...,X be a random sample from Beta(θ,1).
1 n
(9a) (10%) Find a level α union-intersection test of H :θ≦θ≦θ versus
0 1 2
H :θ<θ or θ>θ.
A 1 2
(9b) (10%) Find a (1-α) confidence interval based on inverting the likelihood
ratio test of H :θ=θ versus H :θ>θ.
0 0 A 0
10. (15%) Let T be a statistic with the cumulative distribution function
_
F(t|θ). Moreover, suppose that F(t|θ) and F(t |θ) are non-decreasing
0
function of θ for each t. Construct a (1-α*),α*≦α, confidence interval
for θ by pivoting F(t|θ).
0