Economics 140A
Answers to Exercise 2
1.
a) We are given a sequence
f
Y
i
g
n
i
=1
of independent and identically distributed
random variables with mean
and variance
±
2
. We wish to learn the value of
and to that end construct the estimators
Y
=
1
n
n
X
i
=1
Y
i
and
~
Y
=
1
P
n
i
=1
i
n
X
i
=1
iY
i
:
To verify that the two estimators of
are linear functions of
Y
i
, that is, that the
two estimators are linear estimators, we must verify that each estimator can be
written as
P
n
i
=1
w
i
Y
i
where the weights
f
w
i
g
n
i
=1
do not depend on any element
of the sequence
f
Y
i
g
n
i
=1
. Observe that
Y
=
n
X
i
=1
1
n
Y
i
=
n
X
i
=1
w
i
Y
i
with
w
i
=
1
n
;
and
~
Y
=
n
X
i
=1
w
i
Y
i
with
w
i
=
i
P
n
i
=1
i
=
2
i
n
(
n
+ 1)
;
where the last equality follows from the fact that
P
n
i
=1
i
=
n
(
n
+1)
2
. Because
w
i
does not depend on any element of the sequence
f
Y
i
g
n
i
=1
, both
Y
and
~
Y
are
linear estimators.
b) To verify that the estimators are unbiased, we calculate the expected
value of each estimator. The expected value of
Y
is
E
Y
=
1
n
n
X
i
=1
EY
i
=
so
Y
is an unbiased estimator of
. The expected value of
~
Y
is
E
~
Y
=
1
P
n
i
=1
i
n
X
i
=1
iEY
i
=
1
P
n
i
=1
i
n
X
i
=1
i
=
Because the expected value of both estimators equals the true value of the
parameter, the estimators are unbiased.
c) As both estimators are unbiased, we can directly compare the variances of
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Economics, Variance, Probability theory, WI

Click to edit the document details