Page 298 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 275

tλ
nλe } → e as n → ∞.}
t/n
5.3.1 (Exercise 5.2.1 Continued) Find a and b(> 0) such that we can
claim: as n → ∞.
5.3.2 (Example 5.3.5 Continued) Let X , ..., X be iid Uniform (0, θ)
n
1
with θ > 0 and suppose that T = X , the largest order statistic. First, find
n
n:n
the df of the random variable G = n(θ – T )/T , and then use the Definition
n
n
n
5.3.1 to show directly that the limiting distribution of G is the standard
n
exponential.
5.3.3 Prove CLT using the mgf technique assuming that the X’s are iid having
i
a finite mgf for | t |< h with some h > 0. {Hint: Follow the derivations in Examples
5.3.3-5.3.4 closely.}
5.3.4 (Example 5.3.7 Continued) Under the same conditions as the CLT, along
the lines of Example 5.3.7, derive the asymptotic distribution of
(i) as n → ∞ if µ ≠ 0;
(ii) as n → ∞ if µ > 0 and is positive w.p.1.
In this problem, avoid using the Mann-Wald Theorem.
5.3.5 (Exercise 5.3.2 Continued) Let X , ..., X be iid Uniform (0, θ) with
n
1
θ > 0 and suppose that T = X , the largest order statistic. Does n{θ –
n:n
n
converge to some appropriate random variable in distribution as
n ? ∞? {Hint: Can Slutsky’s Theorem be used here?}
5.3.6 (Exercise 5.2.3 Continued) Let X , ..., X be iid N(µ, 1), –∞ < µ < ∞.
Consider the following cases separately: 1 n

Find suitable a , b (> 0) associated with each T such that
n n n
N (0, 1) as n → ∞. {Hint: Is the Mann-Wald Theorem helpful here?}
5.3.7 (Example 5.3.8 Continued) Find the number η ( > 0) such that
as n → ∞. Here, is the sample variance obtained from iid
random variables X , ..., X , n ≥ 2. Solve this problem separately when the X’s are (i)
i
1
n
normal and (ii) non-normal, under appropriate moment assumptions. {Hint: Is the
Mann-Wald Theorem helpful here?}
5.3.8 (Exercise 5.3.7 Continued) Find the number η( > 0) such that
N(0, η ) as n → ∞, where u (≠ 0) is some fixed
2
real number. Hence, that is without referring to Slutsky’s Theorem, find the
number ξ(> 0) such that as n → ∞ where
u (≠ 0) is some fixed real number. Solve this problem separately when