课程名称︰高等统计推论二
课程性质︰选修
课程教师︰郑明燕
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰102.04.19
考试时限（分钟）：110
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
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)=