Homework 11 – STAT 543
Due Friday, April 21 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday
1. Let X1 , . . . , Xn be iid exponential(θ) and let θ̂n denote the MLE based on X1 , . . . , Xn .
√
(a) Find an asymptotically pivotal quantity based on n(θ̂n − θ).
(b) Find a variance stabilizing transformation (VST) for {θ̂n } and use this to determine a large
sample confidence interval for θ with approximate confidence coefficient 1 − α.
(c) Suppose a random sample X1 , . . . , X100 of n = 100 observations yields x̄n = 1.835464.
Use this information to obtain a large sample confidence interval for θ based on a likelihood
ratio statistic, which has approximate confidence coefficient 90%. (You should be able to
numerically determine the interval.) Using this data, compute also a confidence interval with
approximate confidence coefficient 90% using the VST approach from part(b).
2. For θ > 0, suppose that X1 , . . . , Xn are iid Uniform(0, θ). Consider the MLE of θ, θ̂n = max{X1 , X2 , . . . , Xn }.
d
(a) Prove that, in distribution, n(θ − θ̂n ) −→ Exponential(θ) as n → ∞ (i.e., converges in distribution to an exponential with mean θ).
(Hint: Evaluate Pθ [n(θ − θ̂n ) > t] and remember that lims→∞ (1 + as )s = exp(a).)
(b) Argue carefully that {θ > 0 : θ ≤ nθ̂n /(n − log(20))} can be used as a one-sided (upper)
confidence interval for θ with approximate confidence coefficient 95% (that is, show that the
interval will have an approximate coverage probability of 95% for each θ > 0).
3. Problem 9.21, Casella and Berger (Second Edition)
Some further help regarding notation in the book: X ∼Binomial(n, p), 0 < p < 1, and you are to
show that the Clopper-Pearson confidence interval for p, for each possible value of X = x, is given
by [pL (x), pU (x)] where
pL (x) =
1
1+
n−x+1
F2(n−x+1),2x (1
x
− α/2)
,
pU (x) =
1
1+
n−x
F
(α/2)
x+1 2(n−x),2(x+1)
and Fn1 ,n2 (β) denotes the β-percentile of an F -distribution with n1 , n2 degrees of freedom, i.e.,
P (Fn1 ,n2 ≤ Fn1 ,n2 (β)) = β. When X = 0, we set pL (0) = 0 and when X = n, we set pU (n) = 1
by convention.
Hint: consider a fixed x = 1, . . . , n − 1 value of X; write the binomial probability in terms of
a beta and then in terms of an F using Theorem 5.3.8. It may be useful also that Fn1 ,n2 (β) =
1/Fn2 ,n1 (1 − β) for 0 < β < 1.
4. Problem 9.32, Casella and Berger (Second Edition)
Some further help with notation: the book uses “zα/2 ” to denote “z1−α/2 ” in our standard notation,
where z1−α/2 is the 1 − α/2 percentile of a standard normal Z variable, i.e., P (Z ≤ z1−α/2 ) =
1 − α/2.
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