Stat 5102 (Geyer) Spring 2010
Homework Assignment 3
Due Wednesday, February 10, 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.
31.
Show that the family of Gam(
α,λ
) distributions with
α
known and
λ
unknown, so the parameter space is
{
λ
∈
R
:
λ >
0
}
is a scale family.
32.
Suppose
S
2
n
is the sample variance calculated from an IID normal
random sample of size
n
.
(a) Calculate the bias of
S
n
as an estimator of the population standard
deviation
σ
.
(b) Find the constant
a
such that
aS
n
has the smallest mean square error
as an estimator of
σ
.
33.
Suppose
U
and
V
are statistics that are independent random variables
and both are unbiased estimators of a parameter
θ
. Write var(
U
) =
σ
2
U
and
var(
V
) =
σ
2
V
, and deﬁne another statistic
T
=
aU
+ (1

a
)
V
where
a
is an
arbitrary but known constant.
(a) Show that
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 Spring '03
 Staff
 Normal Distribution, Standard Deviation, Bias of an estimator, moments estimator, asymptotic normal distribution

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