作者air11 (拿出一張白紙...)
看板Statistics
標題[問題] 幾題數統請教
時間Sun Oct 14 11:51:39 2012
小弟我寫台大推甄考古題時，遇到幾個問題，想請各位大大解惑....
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屬於{0.5,1,1.6,1.7,2}
against H_1:θ屬於{2.5,3,6,8,10}
通常我們在做UMP test，虛無假設都是＝、≦、≧這三類，這題是離散型的
我就不太知道怎麼下筆了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))
這題只知道題目要求條件期望值，順序統計量是充分統計量，
不過不知道該怎麼辦......
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
先說這題的函數f(x|θ)是不是有打錯呢?? 指數地方應該是H(θ)+g(x)
才會是指數族吧?!
然後這題感覺照定義直接積分會積不太出來，請問有甚麼比較好的做法呢???
感謝各位大大的指教以及幫忙 :)
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1F：→ IminXD:1.(a)我直覺是畫table先做Neyman-pearson找MP 10/17 02:10
2F：→ IminXD:1.(b)a.做出來後帶帶看點有沒有在規定範圍 10/17 02:11
3F：→ IminXD:2.雙重期望值先下去導導看然後再考慮order stat. ? 10/17 02:15
4F：→ IminXD:4.f(x|θ)和f(x;θ)都是給定θ條件下.指數族的部份應該沒錯 10/17 02:20
5F：→ air11:1.我問過老師，老師說可以想成H0:θ≦2, H1:θ>2，不知道這 10/17 20:26
6F：→ air11:樣行不行....然後1(b)的sample mean是n=180的均數?? 10/17 20:28
7F：→ air11:第二題我不太懂要derive什麼...是要求出甚麼東西嗎?? 10/17 20:29
8F：→ IminXD:n=180,樣本平均數0.3 要你由上面條件導出那串條件期望值 10/17 20:46
9F：→ air11:2依照題目條件，我目前可以想到的只有Rao-Blackwell thm 10/17 21:55
10F：→ air11:但還是沒什麼頭緒QQ... 10/17 21:55