April 4, 2005
Stat 543 Exam 2
Prof. Vardeman
WRITE ALL YOUR ANSWERS ON THIS EXAM!!!
1. It is a Stat 542 probability fact that if X X
1 2
, … , X n
are iid Poisson
( )
, then conditional on i n
=
1
X i
= t , X
1
∼
Bi
⎛
⎝ t ,
1 n
⎞
⎠
. So then in the iid Poisson
( ) ( λ ) exp
( ) =
( )
model, consider estimation of
[
1
=
0 or 1
] a) Identify the UMVUE of this quantity and argue carefully that your estimator is indeed UMVU. b) What is the Cramér-Rao lower bound for the variance of your estimator from a)? Do you expect that this bound is achieved? Why or why not?
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2. Suppose that a discrete random variable X has pmf
θ
. x
(
|
θ )
specified in the table below for some
1 2 3 4 5
θ a) Identify a test
φ ( )
that is MP size
α =
.3
for testing H :
0
θ =
1 vs H :
θ =
2 and carefully state why it has this property. b) Your test from a) could be used to test H :
0
θ =
1 vs H :
θ =
2 or 3 . In this context, is it UMP of its size? Explain carefully. c) Either identify a UMP size
α =
.4
test of H :
0
θ =
1 vs H :
θ =
2 or 4 or argue very carefully that no such test exists.
2

3. Let X be exponential with mean
β
, i.e. have pdf
(
|
β ) = [ >
0
] 1
β exp
⎛
⎝
− x
β
⎞
⎠
on
ℜ
. Let
Y
=
X rounded to the nearest integer be a “digitized” version of X , so that Y has pmf
(
|
β ) =
⎪
⎪
⎧
⎪
⎨
⎪
⎪⎩
⎪
⎪ exp
⎛
⎝
−
⎛
⎝
−
.5
β
⎞
⎠
It is possible to show (don’t try to do so here) that for h h y
(
(
−
β
0 |
0 |
.5
β
β
1
⎞
⎠
2
)
)
−
0 exp
< h h
⎛
⎝
− y
+
β
.5
β
1
< β
2
(
1|
1|
β
β
1
2
( )
)
⎞
⎠ a) Show that the family of distributions specified by y
=
0 y
=
1, 2, 3,
… otherwise
(
|
β )
has MLR in y .
Y . b) Identify a UMP test of size
α =
.2212
of H :
0
β ≥ β
based on the digitized observation
3

4. A curious statistician is interested in the properties of a 5-dimensional distribution on
[ ] 5
that has pdf proportional to p
= exp
⎡
⎣
−
(
( x
1
− x
2
+ x
2
− x
3
) ( x
3
− x
4
+ x
4
− x
5
+ x
5
− x
1
)
2
)
⎤
⎦
This person decides to simulate from this distribution.
Carefully describe EITHER a rejection algorithm that could be used to produce realizations x from this distribution OR a “Gibbs Sampler”/”Successive Substitution Sampling” algorithm that could be used to draw samples from this distribution.
(If you choose to describe a Gibbs sampler, give a formula for the univariate pdf that must be sampled when replacing the i th entry of a current x vector. This formula will involve a 1-dimensional integral that would have to be evaluated numerically.)
4

5. Suppose that X has pdf to test
(
|
θ ) =
2
θ ( − x
) +
2 x on for
θ ∈Θ = [ ]
H :
0
θ ≤
.4 vs H :
θ >
.4
. If the Bayesian’s prior distribution is uniform on
. A Bayesian wants
[ ]
, what is the person’s (0-1 loss optimal) test?
6. Suppose that X is discrete with pmf
( )
be the pmf for and
(
|
θ )
and that is sufficient for
θ ∈Θ ⊂ ℜ
. Let
( )
be the conditional pmf for X given that
( ) = t . We may then write
(
|
θ ) =
(
|
( ) )
Suppose that all useful regularity holds (for example, that
( ( )
|
θ
)
( )
is differentiable in
θ
for all t ).
Argue very carefully that the Fisher information in information in X about
θ
at
θ
(i.e. that
0
I
about
θ
at
θ
is the same as the Fisher
0
( ) =
I
X
( )
).
5