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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