Stat 5102 (Geyer) Spring 2010
Homework Assignment 4
Due Wednesday, February 17, 2010
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
41.
Calculate the ARE of the sample mean
X
n
versus the sample median
e
X
n
as an estimator of the center of symmetry for
(a) The Laplace locationscale family having density given in the brand
name distributions handout.
(b) The
t
(
ν
) locationscale family, with densities given by
f
ν,μ,σ
(
x
) =
1
σ
f
ν
±
x

μ
σ
²
where
f
ν
is the
t
(
ν
) density given in the brand name distributions hand
out. (Be careful to say things that make sense even considering that the
t
(
ν
) distribution does not have moments of all orders. Also
σ
is
not
the
standard deviation.)
(c) The family of distributions called Tri(
μ,λ
) (for triangle) with densities
f
μ,λ
(
x
) =
1
λ
±
1


x

μ

λ
²
,

x

μ

< λ
shown below
±
±
±
±
±
±
±
±
±
±
±
H
H
H
H
H
H
H
H
H
H
H
μ
μ

λ
μ
+
λ
0
1
/λ
The parameter
μ
can be any real number,
λ
must be positive.
42.
Let
X
1
,
X
2
,
...
,
X
n
be an IID sample having the
N
(
μ,σ
2
) distribu
tion, where
μ
and
σ
2
are unknown parameters, and let
S
2
n
denote the sample
variance (deﬁned as usual with
n

1 in the denominator). Suppose
n
= 5
and
S
2
n
= 53
.
3. Give an exact (not asymptotic) 95% conﬁdence interval for
σ
2
.
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 Spring '03
 Staff
 Normal Distribution, Variance, Laplace, Probability theory

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