as094-soln - STAT 854/454 Short Solutions to Assignment 4 1...

STAT 854/454 Short Solutions to Assignment 4
1. Let t
y
=
∑
N
i =1
y
i
be the population total. Consider linear estimators of t
y
in the form
of
ˆ
t
L
=
∑
i ∈ s
w
i
y
i
, where s is the set of sampled units and w
i
is a constant to be used as a
weight for the i th element whenever it is selected for the sample ( i = 1 , 2 , · · · , N ).
(a) Find E (
ˆ
t
L
) and V (
ˆ
t
L
) under a general sampling design using the π
i
and π
ij
.
Soln: Re-write
ˆ
t
L
as
ˆ
t
L
=
∑
N
i =1
a
i
w
i
y
i
, where a
i
= 1 if i ∈ s and a
i
= 0 otherwise. Can
show that E (
ˆ
t
L
) =
∑
N
i =1
π
i
w
i
y
i
and V (
ˆ
t
L
) =
∑
N
i =1
π
i
(1 - π
i
) w
2
i
y
2
i
+2
∑
N - 1
i =1
∑
N
j = i +1
( π
ij
-
π
i
π
j
) w
i
w
j
y
i
y
j
.
(b) Argue that the Horvitz-Thompson estimator is the only unbiased estimator in the class
of linear estimators in the form of
ˆ
t
L
.
Soln: To make E (
ˆ
t
L
) =
∑
N
i =1
π
i
w
i
y
i
= t
y
=
∑
N
i =1
y
i
for an arbitrary y variable, we
must have π
i
w
i
= 1, i.e. w
i
= 1 /π
i
.