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
.

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- Fall '09
- da
- Trigraph, ty
-
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