10/11
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1801 Probability and Statistics: Foundations of Actuarial Science
Assignment 5
Due Date: December 6, 2010
(Hand in your solutions for Questions 2, 6, 8, 11, 12, 14, 18, 22)
1. Based on a random sample of 2 observations, consider two competing estimators of the
population mean
μ
:
2
1
1
2
1
2
1
ˆ
X
X
+
=
,
2
1
2
3
2
3
1
ˆ
X
X
+
=
(a) Are they unbiased?
(b) Which estimator is more efficient? How much more efficient?
2. Let
be a random sample from a distribution with finite mean
n
X
X
X
,...,
,
2
1
and finite
variance
. An estimator of
2
σ
in the form
n
n
X
c
X
c
X
c
L
+
+
+
=
...
2
2
1
1
is called a
linear estimator
, where
are some known constants. If
L
is unbiased,
then it is called a
linear unbiased estimator
. A linear unbiased estimator that has the
minimum variance among all linear unbiased estimators is call the
best linear unbiased
estimator (BLUE)
.
n
a
a
a
,...,
,
2
1
(a) Express
and
in terms of
()
L
E
()
L
Var
,
, and
.
2
n
c
c
c
,...,
,
2
1
(b) Show that the sample mean is the BLUE of
.
3.
An economist gathers a random sample of 500 observations, and loses the records of the last
180. This leaves only 320 observations from which to calculate the sample mean. What is the
efficiency of this, relative to what could have been obtained from the whole sample?
4. Let
{
be a random sample from a distribution with finite mean
}
n
X
X
X
,...,
,
2
1
and finite
variance
. Let
2
(
∑
=
−
−
=
n
i
i
X
X
n
)
1
2
1
1
S
2
be the sample variance. Show that if
()
1
≠
=
S
P
(i.e.
), then the sample standard deviation
S
will always underestimate the
population standard deviation
()
0
≠
S
Var
.
5. Let
{
be a random sample from the normal distribution
}
n
X
X
X
,...,
,
2
1
( )
2
,
σμ
N
. Find the
constant
c
such that
()
∑
=
−
−
=
n
i
i
X
X
n
c
cS
1
2
1
1
is an unbiased estimator of
.
P. 1