課程名稱︰高等統計推論二 課程性質︰數學系選修 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）︰2016/6/23 考試時限（分鐘）：15:30~17:20 試題 : 1. (10%) Let X ,...,X be a random sample from a population with p.d.f f(x) and 1 n c.d.f F(x), which is differentiable. Derive the asymptotic distribution of √n(M -μ), where M and μ are the sample median and population median, n n respectively. 2. (10%) Let X ,...,X be a random sample from Bernoulli(p ) with n≧4. Find 1 n 0 4 the uniformly minimum variance unbiased estimator of p . 0 3. (5%)(10%) Let F (t) and F (c) denote separately the cumulative distribution T C functions of the non-negative continuous random variables T and C. Suppose that T and C are independent and C is noninformative. Based on a random sample of n the form {(X ,δ)} , where X = min{T ,C } and δ= I(X = T ), write down the i i i=1 i i i i i i corresponding likelihood function and derive the maximum likelihood estimator of F (t). T 4. (15%) Let X ,...,X be a random sample from Uniform(θ,θ+1). Consider the 1 n hypotheses H :θ=0 versus H :θ>0. Calculate the power of the unformly most 0 A powerful size α test at any θ > 0. 1 2 2 5. (15%) Let X ,...,X be a random sample from N(μ,σ ), where μ and σ are 1 n unknown. Find the constants a and b so that the (1-α) confidence interval of 2 2 2 2 the form {σ :(n-1)S /b≦σ ≦(n-1)S /a} has the minimum length. n n 6. Let X ,...,X be a random sample from a population with the probability 1 n density function f(x|λ)=λexp(-λx)1 (x), where λ is a positive (0,∞) parameter. (6a) (10%) Find a uniformly most powerful size α, 0<α<1 test for the null hypothesis H :λ=λ versus the alternative hypothesis H :λ>λ , where λ is 0 0 A 0 0 a known constant. (6b) (10%) Find a uniformly most accurate (1-α) confidence interval for λ. 2 7. (5%)(10%) Let X ,...,X be a random sample from N(θ,σ ). Find an unbiased 1 n size α test for the hypotheses H :θ≦θ≦θ versus H :θ<θ or θ>θ. Show 0 1 2 A 1 2 that the given test is unbiased. 8. (15%) Let X ,...,X be a random sample from a Poisson(λ) and λ have a 1 n Gamma(α,β) prior distribution. Find a Bayes test of H :λ≦λ versus 0 0 H :λ>λ. A 0 9. Let X ,...,X be a random sample from Beta(θ,1). 1 n (9a) (10%) Find a level α union-intersection test of H :θ≦θ≦θ versus 0 1 2 H :θ<θ or θ>θ. A 1 2 (9b) (10%) Find a (1-α) confidence interval based on inverting the likelihood ratio test of H :θ=θ versus H :θ>θ. 0 0 A 0 10. (15%) Let T be a statistic with the cumulative distribution function _ F(t|θ). Moreover, suppose that F(t|θ) and F(t |θ) are non-decreasing 0 function of θ for each t. Construct a (1-α*),α*≦α, confidence interval for θ by pivoting F(t|θ). 0 --

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