STAT 543
EXAM II
4/5/06
Name:
Please write clearly as possible and feel free to use the back of the paper if you need more
space. Relax and good luck.
1. Let X1 , . . . , Xn be independent random variables with Xi ∼Poisson(iθ), θ > 0, for i = 1, . . . , n.
(a) Write down the joint probability mass function (pmf) of X1 , . . . , Xn .
P
(b) Show that S = ni=1 Xi is a complete and sufficient statistic for θ.
(NOTE: State any standard result(s) that you are using.)
(c) Show the joint pmf has the monotone likelihood ratio property in S.
(d) Using Method I or otherwise, find the UMVUE of the parametric function γ(θ) = θ2 .
(e) Find the UMVUE of the parametric function γ1 (θ) = e−θ . (Hint: Note that Pθ (X1 = 0) = e−θ .)
(f) Suppose that n = 5 and again X1 , . . . , X5 are independent, Xi ∼Poisson(iθ), θ > 0. Find
the uniformly most powerful (UMP) test of H0 : θ ≥ 1/5 vs H1 : θ < 1/5 of size α = 2e−3 .
(NOTE: You need to determine the numerical values of any constants in the test.)
2. Consider one observation X from the probability density function
µ
f (x|θ) = 1 − θ2 x −
¶
1
,
2
0 ≤ x ≤ 1,
−1 ≤ θ ≤ 1.
For a given α ∈ (0, 1), find the UMP test of size α for testing H0 : θ = 0 vs. H1 : θ 6= 0.
3. Suppose we have two independent random samples: X1 , . . . , Xn are iid N (0, σ 2 ), σ 2 > 0, and
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Y1 , . . . , Yn are iid N (0, τ 2 ), τ 2 > 0. The maximum likelihood estimators (MLEs) are σ̂ 2 = ni=1 Xi2 /n
P
and τ̂ 2 = ni=1 Yi2 /n.
(a) Find the likelihood ratio statistic for testing H0 : σ 2 = τ 2 against H1 : σ 2 6= τ 2 . You may use
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that, under H0 , the MLE is σ̃ 2 = ni=1 (Xi2 + Yi2 )/(2n).
(b) Show that a size α ∈ (0, 1) LRT for testing H0 : σ 2 = τ 2 against H1 : σ 2 6= τ 2 has a rejection
region of the form {T < c1 } ∪ {T > c2 } for some constants c1 and c2 , where T = σ̂ 2 /τ̂ 2 .
(NOTE: You may assume T is continuous and strictly positive. In addition, you may wish to
use the attached graph of h(x) = (1 + x)(1 + x−1 ).)
(c) Write down the condition(s) that c1 and c2 must satisfy for the LRT to have size α.