[試題] 98下 江金倉 統計學 期末考
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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