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、數學研究所、應用數學科學研究所
考試日期︰2015年
考試時限：未知
試題 :
Statistical Inference (Test 1)
～
1 (10%) Let X1,...,Xn be a random sample from a p.d.f f(x|θ0) and θn be an
unbiased estimator for θ0 whcih attains the Cramer-Rao lower bound, where
︿
θ0∈Θ and dim(Θ) = 1. Suppose that the MLE θ of θ0 is a solution of
2 ︿ ～
∂l(θ|X1,...,Xn) = 0 and -∂l(θ|X1,...,Xn) > 0. Show that θn = θn
θ θ
2.(15%) Let X1,...,Xn be a random sample from a population with probability
density function fX (x|θ) = (θ0)x^(θ0 -1), 0 < x <1, 0 < θ0 < ∞.
Derive the asymptotic distribution of the maximum likliehood estimator of θ0.
3. (15%) (5%) Derive the asymptotic normality of the maximum likelihood
estimator under some suitable conditions.
i.i.d
4. (8%) (7%) Let Y1,...,Yn ～ (p0)f(y) + (1-p0)g(y) with p0 being unknwon,
and f(．) and g(．) being knwon p.d.f's. Implement the EM-algorithm to obtain
^(r) ^(r)
an EM-sequecne {p } and show that p will converge to the MLE as r→∞.
n
5. (15%) Let {Xi,δi,Zi}i=1 be a random sample with Xi = min{Ti,Ci},
δi=I(Xi=Ti), and Zi being a p ×1 covariate vector. Suppose that λ(t|z) =
T
(λ0)(t)exp(β0 z) is the hazard function of T conditioning on Z=z. where
(λ0)(t) is a baseline hazard function and β0 is a p ×1 parameter vector.
Conditioning on Z,T and C are further assumed to be independent. Write the
partial likelihood estimation criterion for β.
6. (15%) Let (Y1,x11,...,x1p),...,(Yn,xn1,...,xnp) be independent with
E [Yi|xi1,...,xip] = (mi)π0(xi1,...,xip) and Var(Yi|xi1,...,xip) =
π0 π0
ψ*mi*π0(xi1,...,xip)*(1-π0(xi1,...,xip)), i=1,...,n, where xi1,...,xip are
cobariates, π0(xi1,...xip) = exp(Ａ)/(1+exp(Ａ)), where
Ａ = β00 + β01*xi1 + ... + β0p*xip,
and ψ is a scale parameter. Show that the quasi-score estimator is different
from the least squares estimator for (β00,β01,...,β0p).
7. (10%) Suppose that Yi～Xυi, i=1,...,k, are mutually independent. Try to
k
find the Satterthwaite approximation for Σ ai*Yi, where ai's are known
i=1
constant.