課程名稱︰高等統計推論
課程性質︰選修∕數研所統計組必修
課程教師︰陳宏 教授
開課系所︰數學系
考試時間︰2006/06/21 13:20-15:20
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試題： Advanced Statistical Inference II
Final Test (Spring 2006)
1. (60 points) Let X_i, i＝1,2, be independent with p.d.f.'s
λ_i‧exp(-λ_i‧x)‧I_(0,∞)(x), i＝1,2, respectively. Let θ＝λ_1∕λ_2.
Show that θX_1∕X_2 is a pivotal quantity and construct a confidence
interval for θ with confidence coefficient 1－α, using the pivotal
quantity.
2. (100 points) Let X_1,...,X_n be iid from the uniform distribution U(0,θ)
with an unknown θ＞0.
-1 -1
(a) (40 points) Consider the interval of the form [b X_(n), a X_(n)]
-n -n
where b －a ＝1－α. Determine a and b to make an interval with
shortest length.
(b) (10 points) Find the distribution of ln(X_1).
~ n 1/n
(c) (20 points) Find the asymptotic distribution of X＝(Π X_i) .
i=1
(d) (20 points) Use (c) to propose a confidence interval of θ.
(e) (10 points) For the intervals obtained in (a) and (d), which one do you
prefer? Give reasons to support your claim.
3. (100 points) Let X_1,...,X_k be independent and Poisson distributed with
parameter λ_i as x_i
λ_i -λ_i
f(x_i;λ_i)＝───‧e , x_i＝0,1,...
x_i!
where λ_i＞0 are unknown.
(a) (10 points) Show that the MLE of λ＝(λ_1,...,λ_k) is
︿ ︿ ︿
λ＝(λ_1,...,λ_k)＝(X_1,...,X_k).
(b) (10 points) Consider the composite hypothesis H_0: λ_i＝αd_i,
i＝1,...,k, where d_i are known numbers and λ＞0 is unknown. Derive
︿ ︿
the MLE λ_10,...,λ_k0 under this hypothesis.
(c) (30 points) Find the Wald test statistic for H_0.
(d) (30 points) Find the LRT test under H_0 versus H_2: λ_i≠αd_i and
indicate its asymptotic distribution.
(e) (20 points) Find the LRT test under H*_0: λ_i＝d_i, i＝1,...,k versus
H_0. Please describe its asymptotic distribution.
4. (50 points) Suppose that X_1, X_2 is a random sample from a uniform
distribution over (0,1).
(a) (30 points) Find the probability of the random interval (X_1∕(3X_2),
2X_1∕X_2) containing 2/3.
(b) (20 points) Among all random intervals of the form (aX_1∕X_2, bX_1∕X_2)
where 0≦a＜b, find the shortest interval containing 2/3 with
probability 7/12.
2
5. (50 points) Observe (x_1, Y_1),...,(x_n, Y_n) where Y_i～N(α＋βx_i, σ )
n
and Σ x_i＝0.
i=1
(a) (20 points) Derive the maximum likelihood estimators of α and β.
(b) (30 points) Construct a 95% confidence interval for α＋β.
Remark: For questions 4 and 5, you only need to answer one only.
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