作者air11 (拿出一张白纸...)
看板Statistics
标题[问题] 几题数统请教
时间Sun Oct 14 11:51:39 2012
小弟我写台大推甄考古题时，遇到几个问题，想请各位大大解惑....
1. Suppose that X_1, X_2, ..., X_n is random sample from Gamma(v,θ),
where v>0 is some known constant and θ>0 is an unknown parameter
(a) Construct a uniformly most powerful test, with significance level
α, for testing the hypothesis H_o:θ属於{0.5,1,1.6,1.7,2}
against H_1:θ属於{2.5,3,6,8,10}
通常我们在做UMP test，虚无假设都是＝、≦、≧这三类，这题是离散型的
我就不太知道怎麽下笔了QQ
(b) Suppose that v=0.2 and there is a sample of size 180 with sample mean
0.3 . Under significance levelα=0.05, does the test in (a) reject
H_0 or not?
2. Let X_1, X_2, ...,X_n be a random sample with E(h(X_1,X_2))=θ, where
h(x,y) is a symmetric function. Moreover, let X_(1),...,X_(n) denote
the order statistics of X_1, X_2, ...,X_n. Derive the conditional
expection E(h(X_1,X_2)|X_(1),...X_(n))
这题只知道题目要求条件期望值，顺序统计量是充分统计量，
不过不知道该怎麽办......
3. Let I(f)=∫f(x)dx and X_1,...,X_n be a random sample from a density
a
function g(x) on [a,b]. Find an unbiased estimator of I(f) & compute
it's variance
4. Let X_1, ...,X_n be a random sample from a one parameter exponential
family f(x|θ)=exp(θh(x)-H(θ)g(x)), where H'(θ)=h(θ) and h'(θ)>0
(a) Show that E(h(X)|θ)=h(θ) and Var(h(X)|θ)=h'(θ)
(b) Find the uniformly most powerful level α test of
H_0:θ≦θ_0 vs. H_1:θ>θ_0
先说这题的函数f(x|θ)是不是有打错呢?? 指数地方应该是H(θ)+g(x)
才会是指数族吧?!
然後这题感觉照定义直接积分会积不太出来，请问有甚麽比较好的做法呢???
感谢各位大大的指教以及帮忙 :)
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1F：→ IminXD:1.(a)我直觉是画table先做Neyman-pearson找MP 10/17 02:10
2F：→ IminXD:1.(b)a.做出来後带带看点有没有在规定范围 10/17 02:11
3F：→ IminXD:2.双重期望值先下去导导看然後再考虑order stat. ? 10/17 02:15
4F：→ IminXD:4.f(x|θ)和f(x;θ)都是给定θ条件下.指数族的部份应该没错 10/17 02:20
5F：→ air11:1.我问过老师，老师说可以想成H0:θ≦2, H1:θ>2，不知道这 10/17 20:26
6F：→ air11:样行不行....然後1(b)的sample mean是n=180的均数?? 10/17 20:28
7F：→ air11:第二题我不太懂要derive什麽...是要求出甚麽东西吗?? 10/17 20:29
8F：→ IminXD:n=180,样本平均数0.3 要你由上面条件导出那串条件期望值 10/17 20:46
9F：→ air11:2依照题目条件，我目前可以想到的只有Rao-Blackwell thm 10/17 21:55
10F：→ air11:但还是没什麽头绪QQ... 10/17 21:55