課程名稱︰高等統計推論二 課程性質︰應數所數統組必修 課程教師︰江金倉 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/6/16 考試時限（分鐘）：16:30~17:20 試題 : 1. (15%) Consider the hypotheses H :θ∈Θ versus H :θ∈Θ. Let W(X ,...,X ) 0 0 A 1 1 n be a test statistic such that large values of W(X ,...,X ) give evidence that 1 n H is ture. Define p(x ,...,x )=sup P(W(X ,...,X )≧W(x ,...,x )|θ). Show A 1 n θ∈Θ 1 n 1 n 0 that p(X ,...,X ) is a valid p-value. 1 n 2. (20%) Let X ,...,X be a random sample from a p.d.f f(x|θ) with θ∈Θ and 1 n dim(Θ)=k. Consider the null hypothesis H :θ∈Θ versus the alternative 0 0 hypothesis H :θ∈Θ-Θ, where Θ={θ:θ=g(η)}⊂Θ with η is a (k-r) ×1 A 0 0 unknown parameter vector and g(‧) is a continuously differential function from k-r k d 2 R to R . Show that -2ln(λ )→χ under H and the regularity conditions, n r 0 where λ is the likelihood ratio test statistic. n 2 2 3. (15%) Let X ,...,X be a random sample from N(μ,σ) with σ being unknown. 1 n Derive the power function of the size α likelihood ratio test for the null hypothesis H :μ≦μ versus the alternative hypothesis H :μ>μ with the power 0 0 A 0 function being expressed via Φ (‧), which is the t distribution with ν T,ν degrees of freedom. 4. (10%)(10%) Let p (X ,...,X ) denote a valid p-value for H :θ∈Θ versus j 1 n 0j j H :θ not in Θ, j=1,...,k. Define a valid p-value p(X ,...,X ) for the Aj j 1 n k k c hypotheses H :θ∈∪ Θ versus H :θ∈∩ Θ and find a level α test based on 0 j=1 j A j=1 j p(X ,...,X ). 1 n 5. (10%) Let X ,...,X be a random sample from Uniform(θ,θ+1). Find the 1 n uniformly most powerful size α test of the hypotheses H :θ=0 versus H :θ>0. 0 A 6. (10%)(10%) Let X ,...,X be a random sample from Bernoulli(p ) and Y ,...,Y 1 n 1 1 m be another independent random sample from Bernoulli(p ). A hypothesis test of 2 interest is H :p = p versus H :p ≠p . Find a Wald test statistic and a score 0 1 2 A 1 2 test statistic. --

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