課程名稱︰高等統計推論二
課程性質︰選修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰102.04.19
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Advanced Statistical Inference II Midterm Examination 19 Apr 2014
1. A company wants to estimate the proportioin p, 0＜p＜1, of dedective items
it produces. Since they rarely produce defective items, n workers are asked
to continue inspecting until they each has observed one defective item. For
th
i=1,2,....,n, let X be the number of items the i worker inspect before
s/he has observed one defective item.
(a)5% What assumptions are needed so that the model X1,......,Xn i.i.d.
distributed as Geometric(p) is appropriate for this setting?
^ n ^
(b)10% Let p1= (n-1)(Σ Xi-1)^(-1). Show that p1 is the uniformly minimum
i=1
variance unbiased estimator of p.
(c)10% ^
Show that the maximum likelihood estimator of p, denoted as p2, is biased.
2. Suppose that X1,X2,....,Xn is an i.i.d sample from the Uniform(-θ,θ)
distribution which has density f(x)=2^(-1)θ^(-1)I (x), where I is
(-θ,θ)
the indicator funtion.
^
(a)8% Show that the maximum likelihood estimator of θ, denoted as θ1 is ^
biased and has a simple rescaling which is unbiased. Call this rescaling θ2.
^
(b)8% Is θ2 an uniformly minimum variance unbiased estimator of θ ?
Why or why not?
3. A random sample X=(X1,....,Xn) is drawn from a Pareto population with pdf
θυ^θ
f(x│θ,υ)=