課程名稱︰高等統計推論下 課程性質︰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）︰99/6/24 考試時限（分鐘）：180 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Let X_1,...,X_n be arandom sample from a p.d.f. f(x|θ)=H(x)C(θ)exp(θT(x)) where the range of f_X(x|θ) is independent of θ and C(θ) is differentiable Find the UMVUE of E(T(X)|θ). 2. Suppose that X_1,...,X_n be a random from a Poisson(λ). (2a) Find the UMVUE of exp(λ). (2b) Calculate the asymptotic relative of the UMVUE with respect to the MLE n 3. Let X_1,...,X_n, be a random sample from a Bernoulli(θ), T_n=((1/n)ΣX_i)^2 n i=1 and T_nj=((1/(n-1))ΣX_i)^2, j=1,...n. ~ i≠j n Show that T_n=n*T_n-((n-1)/n)ΣT_ni is the UMVUE of θ^2. i=1 4. Let X_1,...,X_n be a random sample from a uniform(θ,θ+1) with X_(1),...,X_(n) being order statistic. (4a) Show that the test γ with rejection region R(X_1,...,X_n) = {(X_1,...,X_n):X_(n)＞1 or X_(1)≧1-α^(1/n)} is the UMP size α for the hypothesis H_0:θ=0 versus H_A:θ＞0. (4b) Calculate the power of the test γ at θ, θ＞0. 5. Let X_1,...,X_n be a random sample from N(μ,σ^2), σ^2 is unknown. Define a valid p-value for the hypothesis H_0:μ=μ_0 versus H_A:μ≠μ_0. 6. Let X_1,...,X_n be a random sample from Poisson(λ) and λ have a Gamma(α,β) prior distribution. Establish a Bayesian test for the hypothesis H_0:λ≦λ_0 versus H_A:λ＞λ_0. 7. Let X_1,...,X_n be a random sample from a U(0,θ). Construct the shortest (1-α) pivotal interval for θ based on the pivotal quantity X_(n)/θ. 8. Let X_1,...,X_n be a random sample from Poisson(λ). Find the uniform most accurate (1-α) lower sided confidence interval for λ. 9. Let X_1,...,X_n be a random sample from a p.d.f. f(x|θ) and suppose that {θ:L(X_1,...,X_n)≦θ≦U(X_1,...,X_n)} is a most accurate (1-α) confidence interval for θ. Find a UMA (1-α) confidence interval for 1/θ 10.Let f(x_1,...,x_n|θ) be a conditional p.d.f. of a random sample X_1,...,X_n , and l(θ,C)=b*length(C)-I_{C}(θ), where b is a nonegative constant and C C is a credible set of θ which minimizes the Bayes risk. 11.Let X_1,...,X_n be a random sample from a Uniform(θ_1,θ_2). Show that (X_(1),X_(n)) is the minimal sufficient statistic of (θ_1,θ_2). --

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