這是台大推甄的考古題，小弟我碰到幾題想不到怎麼解，請各位大大幫忙
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......
b
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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