課程名稱︰高等統計推論一 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）︰2016/1/14 考試時限（分鐘）：15:30~17:20 試題 : 1. (10%) Let T be a continuous failure time with the survival function S(t) = P(T > t), 0 < t < ∞, and finite first moment. Express the expected residual life r(t) = E[T-t|T≧t] at time t in terms of a functional of S(t). 2. (10%) Let T ,...,T be a random sample from an exponential iid distribution, 1 n and T ,...,T be the corresponding order statistics. Derive the joint (1) (n) distribution of U ,...,U , where U = (n-i+1)(T -T ), i=1,...n, 1 n i (i) (i-1) with T = 0. (0) 3. (25%) Suppose that {N (t):t∈[0,τ]} is a nonhomogeneous Poisson process 1 with rate function λ(t) > 0 and {N (t)} is a Poisson process with rate 2 λ=1. Moreover, let M = N(τ) and T ,..., T be the corresponding recurrent 1 M event times of {N (t):t∈[0,τ]}. 1 (3a) (10%) Derive the joint distribution of T ,..., T given M = m. 1 M (3b) (7%)(8%) Generate M and T ,..., T through {N (t)}. 1 M 2 4. (4%)(4%)(6%)(6%) State and show the weak law of large numbers and central limit theorem. 5. (10%) Let X ,..., X be a random sample from Bernoulli(p) with p≠0.5 and 1 n _ n _ _ X = Σ X /n. Derive the asymptotic distribution of X (1-X ). n i=1 i n n 2 6. (15%) Let X ,..., X be a random sample from N(θ,θ ) with θ > 0, 1 n _ n 2 n _ 2 X = Σ X /n, S = Σ(X - X) /(n-1), and E[cS|θ]=θ for some constant c. Find i=1 i i=1 i _ the value of k∈[0,1] such that the statistic kX + (1-k)(cS) has the minimin variance. 7. (10%) Let X~f(x) and {Y ,..., Y } be a random sample from g(y), which has 1 n * the same support of f(x), and P(X = Y ) = q with k k n q = (f(Y )/g(Y ))/(Σ f(Y )/g(Y )), k = 1,...,n. Find a function q(x) such k k k i=1 i i * p that P(X ≦x) → q(x) as n→∞. n -n k n n-1 -u 8. (5%)(5%) Compute the limits of Σ e n /k! and ∫u e du/Γ(n) as n→∞. k=0 0 9. (15%) Let X ,..., X be a random sample from a population with probability 1 n density function f(x) and differentiable cumulative distribution function F(x). Derive the limiting distribution of (√n)(M - μ), where M and μ are n n separately the sample median and population median. --

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