Problem Set 7
ECON 837
Prof. Simon Woodcock, Spring 2006
Due: Wednesday April 5
1. (Casella and Berger, 1990) Suppose that
X
1
; X
2
; :::; X
n
are an iid random sample from
a Bernoulli
(
p
)
distribution. That is,
f
(
x
j
p
) = Pr [
X
=
x
j
p
] =
p
x
(1
°
p
)
1
°
x
x
2 f
0
;
1
g
; p
2
[0
;
1]
:
Let
°
x
=
1
n
P
n
i
=1
x
i
:
(a) Show that
°
x
is an unbiased estimator of
p:
(b) Show that the variance of
°
x
attains the CramerRao lower bound, so that
°
x
is the
best unbiased estimator of
p:
2. [Final Exam, 2004] Let
y
=
°
i
+
"
where
i
is an
n
vector of ones and
"
±
N
(
0
;
I
)
:
(a) What is the MLE of
°
?
What is its asymptotic distribution?
(b) Derive the likelihood ratio statistic (and give its asymptotic distribution) for test
ing
H
0
:
°
= 0
; H
1
:
°
6
= 0
:
(c) Derive the Wald test statistic (and give its asymptotic distribution) for testing
the same hypothesis.
(d) Derive the score (LM) test statistic (and give its asymptotic distribution) for
testing the same hypothesis.
(e) Suppose you are testing °at the 0.05 level.±Sketch the general shape of the func
tion
Pr [
reject
H
0
j
°
]
as a function of
°
. Indicate how the shape of this function
changes as the sample size increases.
3. [Final Exam, 2005] Suppose you are interested in the regression model
y
=
X
°
+
"
where
y
is
N
²
1
;
X
is
N
²
k;
°
is
k
²
1
;
and
"
is
N
²
1
:
Assume that
"
±
N
(0
; ±
2
I
n
)
,
that
y
and
X
are both measured in deviations from means, and the model has no
intercept. You are interested in testing the hypothesis
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 Spring '09
 Thewalt
 Null hypothesis, Statistical tests, Likelihoodratio test, Statistical theory, GLS

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