Homework 10 – STAT 543
Due Friday, April 14 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday
1. Consider one observation X from the probability density function
µ
¶
1
2
f (x|θ) = 1 − θ x −
, 0 ≤ x ≤ 1,
−1 ≤ θ ≤ 1.
2
Suppose we wish to test H0 : θ = 0 vs. H1 : θ 6= 0.
(a) For the above hypotheses, find the likelihood ratio test statistic λ(X) based on X (i.e., as a
function of X)
(b) For a given α ∈ (0, 1), find the size α likelihood ratio test (LRT) of the above hypotheses.
(Hint: Consider α < 1/2 or ≥ 1/2.)
2. Problem 9.4, Casella and Berger (2nd Edition)
3. Problem 9.10(b), Casella and Berger (2nd Edition) (We did part (a) in class.)
4. Problem 9.11, Casella and Berger (2nd Edition)
5. Problem 9.13, Casella and Berger (2nd Edition)
6. Problem 9.16, Casella and Berger (2nd Edition)
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