Homework 12 – STAT 543
Not to turn in - practice only
1. Let X1 , . . . , Xn be iid random variables with the common cdf
½
0
if x ≤ θ
F (x, θ) =
−3
1 − (x/θ)
if x ≥ θ,
where θ > 0.
(a) Find the cdf of Tn ≡ X(1) /θ where X(1) denotes the minimum of X1 , . . . , Xn .
(b) Find a (1 − α) 2-sided confidence interval for θ based on the pivotal quantity Tn .
(c) Find a size α LRT for testing H0 : θ = θ0 against H1 : θ 6= θ0 , where θ0 > 0 is fixed.
(d) Invert the test in part (c) to get a 2-sided 1 − α confidence interval for θ.
2. 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.
3. 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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