課程名稱︰統計學 課程性質︰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）︰2010.06.21 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. (10%)(10%) Let X_1,...,X_n be a random sample from a distribution with the density function f(x│θ) = 0.5e^(-︱x-θ︱) I_{(-∞,∞)}(X). Find the maximum likelihood estimastor of θ and derive its sampling distribution. 2. Let X_1,...,X_n be a random sample from a population with probability density function fx(x│θ) = θx^(θ-1), 0<x<1, 0<θ<∞. (2a) (10%) Find the uniformly minimum variance unbiased estimator of θ. (2b) (10%) Write the asymptotic distribution of the maximum likelihood estimator of θ. 3. (15%) Let X_1,..., X_n be a ramdom sample from a normal distribution with mean μ and variance σ^2. Find the κth moment of n _ _ n Sn^2 = Σ (X_i - X_n)^2 / (n-1), κ = 1, 2, ..., where X_n = n^(-1)Σ X_i. i=1 i=1 4. (15%) Let X_1, ..., X_n be a random sample from a continuous distribution F(x) with the correstponding order statistics X_(1), ..., X_(n). Compute the expectation of F(X_(i)), i=1, ..., n. 5. (15%) Let X_1, ..., X_n be a random sample from Poisson(λ) and λ have Gamma(α,β), where α and β are known positive constants. Find the Bayes estimator of λ under the squared error loss function. 6. (15%) Let X_1, ..., X_n be a random sample from a Poisson distribution with rate λ. Derive a likelihood ratio test of H_0 :λ=λ_0 versus H_A: λ=λ_1 (λ_1 > λ_0) at level α, and show that the rejection region n is of the form {(X_1, ..., X_n) :Σ X_i > c}, where c is the smallest i=1 n constant satisfying P(Σ X_i > c│λ_0)≦α. i=1 7. Suppose that X_1, ..., X_m are independent with X_i~ Binomial(n,p_i). (7a)(15%) Derive a likelihood ratio test for the hypothesis H_0 : p_1 = ... = p_m versus the alternative hypothesis H_A : p_i ≠p_j for some i≠j. (7b)(10%) What is the large sample distribution of the test statistic? --

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