Unformatted text preview: 36226 Summer 2010 Quiz 3 July 23 1. (2 points) What is a pivotal quantity?
A pivotal quantity satisﬁes the following two points:
• It is a function of the data and the unknown parameter(s)
• It has a distribution which does not depend on any unknown parameters
2. (1 point each) Let X ∼ exp(θ) and let Y1 , . . . , Yn be iid exp(θ). Which of the following are
pivotal quantities? You may use the following facts:
1
• fX (x) = e−x/θ
θ
• FX (x) = 1 − e−x/θ
2
X ∼ χ2 (2)
θ
2n
•
Y ∼ χ2 (2n)
θ • (a) 1 − e−X/θ
Yes
(b) 2Y
No
Y −θ
√ if n is large.
(c)
s/ n
Yes
3. (5 points) Let X ∼ exp(θ). Consider the hypothesis test H0 : θ = 1/5 versus Ha : θ = 1/2.
A size α = 0.05 test for this hypothesis is X in the rejection region R = [− 1 ln 0.05, ∞), i.e.
5
1
reject the null if we observe X greater than c = − 5 ln 0.05. Calculate the probabilities of
Type I and Type II error. Notice that the PDF and CDF of X are given in problem 2. P(Type I error) = PH0 (X ∈ R) = α = 0.05.
P(Type II error) = PHa (X ∈ R)
= FX (c  θ = 1/2) = 1 − e−2c
=1−e 2
5 ln 0.05 = 1 − (0.05)2/5 ≈ 0.70 1 (you can stop here) ...
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This document was uploaded on 07/14/2011.
 Summer '09

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