ORIE 6700 MIDTERM, FALL 2010
Rules:
Use only class notes and the one and only course text.
You must work on your own without
collaborating. Giving help is as serious a form of cheating as requesting help.
(1) Let
X
1
, . . . , X
n
be a random sample from
f
(
x

θ
) =
θ
2
xe

θx
1
(0
,
∞
)
(
x
)
where
θ >
0.
(a) Name this density.
(b) Compute
I
(
θ
) =
E

∂
2
∂θ
2
log
f
(
X
i

θ
) .
(2) Suppose that
X
i
iid
∼
N
(
μ,
1) for
i
= 1
, . . . , n
where 0
≤
μ
≤
1. Suppose that we are estimating
μ
with
squared error as the loss function.
(a) Find the risk functions for
T
1
=
¯
X
,
T
2
=
¯
X/
2, and
T
3
= 1
/
4 +
¯
X/
2.
(b) Calculate the Bayes risk of
T
3
under the prior
π
(
μ
) =
1
{
μ
∈
[0
,
1]
}
(uniform on the range 0 to 1).
(3) Suppose that
X
i
iid
∼
Discrete(
θ,
2
θ,
3
θ,
(1

6
θ
)) for
i
= 1
, . . . , n
where 0
< θ <
1
/
6.
(a) Find
I
(
θ
) =
E

∂
2
∂θ
2
log
f
(
X
i

θ
)
(b) What is the CramerRao bound for for the variance of unbiased estimators of
θ
, the probability
of observing the first outcome?
(4) (Poisson regression) Let
Z
1
, . . . , Z
n
be fixed constants and let
X
1
, . . . , X
n
be independent Poisson
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 Fall '10
 WOODARD
 Normal Distribution, Probability theory, Estimation theory, Bayes risk

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