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 --

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