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.