課程名稱︰高等統計推論二
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）︰2016/3/28
考試時限（分鐘）：11:20~12:10
試題 :
1. (8%)(7%) 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 maximum
0 0 0 0 0 0
likelihood estimator and moment estimator of (α,β).
0 0
2. (13%) Let X ,...,X be a random sample from a population with probability
1 n
θ_0
θν
0 0
density function f(x|θ,ν)=—————1 (x), where θ and ν are unknown
0 0 θ+1 [ν,∞) 0 0
0 0
x
positive parameters. Find the maximum likelihood estimators of θ and ν.
0 0
iid
3. (10%)(7%) Let Y ,...,Y ~ p f(y)+(1-p )g(y) with p being unknown, and f(‧)
1 n 0 0 0
and g(‧) being known p.d.f's. Implement the EM-algorithm to obtain an
^(r) ^(r)
EM-sequence {p } and show that p will converge to the MLE as r→∞.
n
4. (15%) Let {X ,δ ,Z } be a random sample with X = min{T ,C }, δ=I(X =T ),
i i i i=1 i i i i i i
T
and Z being a p ×1 covariate vector. Suppose that λ(t|z)=λ(t)exp(β z) is
i 0 0
the hazard function of T conditioning on Z=z, where λ(t) is a baseline hazard
0
function and β is a p ×1 parametr vector. Conditioning on Z, T and C are
0
further assumed to be independent. Write the partial likelihood estimation
criterion for β.
5. (13%) Let n ,...,n be the frequencies of a random sample from Gamma(α,β)
1 k 0 0
2
in the classes χ,...,χ. Find the minimum χ estimator of (α,β).
1 k 0 0
6. (6%)(6%) Define the class of exponential dispersion models and write the
corresponding score function.
7. (15%) Let (Y ,x ,...,x ),...,(Y ,x ,...,x ) be independent with
1 11 1p n n1 np
E [Y |x ,...,x ] = m π(x ,...,x ) and Var (Y |x ,...,x )
π i i1 ip i 0 i1 ip π i i1 ip
0 0
= ψm π(x ,...,x )(1-π(x ,...,x )), i=1,...,n, where x ,...,x are
i 0 i1 ip 0 i1 ip i1 ip
covariates, π(x ,...,x ) = exp(β +β x +...+β x )/(1+exp(β +β x +
0 i1 ip 00 01 i1 0p ip 00 01 i1
...+β x )), and φ is a scale parameter. Show that the quasi-score estimator
0p ip
is different from the least squares estimator for (β ,β ,...,β ).
00 01 0p