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February 25, 2009
Name
Student ID
The exam is closed book and closed notes. You may use one 8
1
2
×
11
sheet of paper with formulas, etc. You may also use the handouts on “brand
name distributions” and Greek letters. Put all of your work on this test form
(use the back if necessary). Show your work or give an explanation of your
answer. No credit for numbers with no indication of where they came from.
The points for the questions total to 100. There are 5 pages and 5 prob
lems.
1. [20 pts.] The function
f
μ
(
x
) =
2
π
(
e
x

μ
+
e

(
x

μ
)
)
,
∞
< x <
∞
is a probability density function (PDF), where the parameter
μ
can be
any real number. The mean and variance of this distribution are
E
(
X
) =
μ
var(
X
) =
π
2
4
You do not have to prove any of the above. Given that information, ﬁnd
the asymptotic relative eﬃciency (ARE) of the sample mean and sample
median of an independent and identically distributed (IID) sample from
this distribution, both considered as estimators of
μ
.
1
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 Spring '09
 Geyer

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