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︰2006/04/19 13:20～15:30
試題： Advanced Statistical Inference II
Midterm (Spring 2006)
1. Let X_1,...,X_n be i.i.d. from a Poisson distribution P(θ) with an
unknown θ＞0.
(a) (30 points) Find the maximum likelihood estimate of η＝exp(-θ).
_
nX
(b) (30 points) Let T_n＝(1－1/n) . Find the bias and mean square error
of T_n as an estimator of η.
(c) (40 points) Determine the asymptotic behavior of T_n. Discuss on its
consistency and asymptotic distribution.
2. Suppose Y_1,...,Y_n arise from the following AR(1) model:
Y_j＝μ＋ρ(Y_j-1－μ)＋ε_j, j＝1,2,...
iid 2 2
where ε_j ～N(0,σ ), μ屬於Ｒ, σ ＞0 and -1＜ρ＜1. Suppose that
Y_0＝y_0 where y_0 is a given constant.
(a) (20 points) Write down the density function of Y_1 and the conditional
density function of Y_2 given Y_1＝y_1
(b) (20 points) Write down the joint density function of (Y_1,Y_2).
2
(c) (30 points) Write down the likelihood function of θ＝(μ,σ ,ρ) and
find the minimal sufficient statistic for θ.
(d) (30 points) Propose a method of finding MLE in (c).
3. Suppose that X_1,...,X_n is a random sample from a normal distribution
2
N(μ,μ ), where μ＞0.
︿
(a) (30 points) Show that the likelihood equation has a unique root μ
within the parameter space unless X_i are all equal to zero.
︿ 2
(b) (30 points) Show that the asymptotic variance of μ is equal to μ∕(3n).
You may without proof assume that the necessary regularity conditions
are satisfied.
(c) (40 points) Consider the estimator
～ _ ___________
μ＝{X＋2√SSD∕(n－1)}∕3,
n _ 2 ～
where SSD＝nΣ(X_i－X) . Derive the asymptotic distribution of μ.
i=1
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