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Unformatted text preview: 36226 Summer 2010 Quiz 4 July 30 1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement.
(a) If I fail to reject the null hypothesis with α = 0.01 then I can reject the null with
α = 0.05.
FALSE
(b) The pvalue is the probability that the alternative hypothesis is true given the data I
have seen.
FALSE
(c) If {X > 2} is the rejection region for an MP test of the hypothesis H0 : θ = 0 versus
Ha : θ = 3, then this is also the rejection region for a UMP test of the hypothesis
H0 : θ = 0 versus Ha : θ > 0.
TRUE
(d) Let X ∼ N(µ, 1). There is no UMP test for the hypothesis H0 : µ = 0 versus Ha : µ = 0.
TRUE
(e) If I fail to reject the null hypothesis, then the pvalue was greater than α.
TRUE
2. Let X1 , . . . , Xn be i.i.d. U(0, θ). Show that the Pareto distribution with density
p(θ) = aba
I
(θ)
θa+1 [b,∞) is conjugate and ﬁnd the new parameters a and b of the posterior distribution.
Hint: Remember to keep around EVERYTHING that depends on θ when you proportion
things away.
Hint 2: Think about indicator functions as logicals and remember that in the posterior, X
is ﬁxed and θ is random.
1
I (X )
θn [0,θ] (n)
1
p(θ) ∝ a+1 I[b,∞) (θ)
θ
1
1
p(θ  X1 , . . . , Xn ) ∝ n I[0,θ] (X(n) ) a+1 I[b,∞) (θ)
θ
θ
1
∝ n+a+1 I[max{b,X(n) },∞) (θ)
θ
∝ Pareto(a = a + n, b = max{b, X(n) })
L(θ) = So the Pareto is conjugate and the posterior has new parameters a = a + n and b =
max{b, X(n) }. 1 ...
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This document was uploaded on 07/14/2011.
 Summer '09

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