146
B. Panahbehagh and R. Bruggemann
Table 6 Example of POC and POR for a set with 5 units with R = 2. According to Poset, there is
an order between all the units except b that is incomparable with c, d and e. For these units there
are 4 LEs which leads to an unequal size post-stratified initial sample. For these units, 4-th stratum
will be empty and instead we will have two units in 3-th stratum. In POC we use mean result of
the all LEs and for POR we just select one of them randomly
POC
POR
Set X
Y
LE1 LE2 LE3 LE4 Set MH Stratum LE3 Set Stratum
e
8
7
e
e
e
b
e
4.75 5
e
e
5
d
4
6
d
d
b
e
d
3.50 3
b
d
3
c
3
5
c
b
d
d
c
2.25 2
d
c
2
b
1
9
b
c
c
c
b
3.50 3
c
b
4
a
0
0
a
a
a
a
a
1.00 1
a
a
1
Table 7 General POC. Please note that Z {h}i = (X {h}i , X {h}i ) is a unit that has been fallen into
the h-th stratum after i − 1 units, according to its mean height MH in respective LEs
Ranks
1 (1-th Stratum) 2 (2-th Stratum) · · · m (m-th Stratum)
Z {1}1
Z {2}1
· · · Z {m}1
Z {1}2
Z {2}2
· · · Z {m}2
. . .
. . .
. . .
. . .
. . .
. . .
· · · Z {m}Km
Z {1}K 1
. . .
Z {2}K 2
to POC (see Table 7). Panahbehagh (2020) showed that we can estimate unbiasedly
the elements of µ by
µ POC = ( μ x.POC , μ y.POC ) =
m
h=1
W h ¯
Z {h} , W h =
K h
Km
, ¯
Z {h} =
1
d h
iis h
Z {h}i .
(2)
where Z {h}i = (X {h}i , Y {h}i ) is a unit that has been fallen into the h-th stratum after
i − 1 units, according to its MH in the respective LEs.
For POC, because K h ; h = 1, 2, . . . , m are random, with unknown distribution,
mean and variance it would be complicated to calculate the variance of µ POC .
Then it is better to present a design with equal size post stratified-initial sample.
For this purpose, POR is presented. In POR, just one of the LEs would be selected
to decide about putting the units in the strata. In example of Table 6, assume LE3
is selected, then we will put the units of the set in the strata according to ranks of
them in LE3. With repeating this procedure for K sets we will have a post-stratified
initial sample with equal size. Now with showing the sample unit in h-th stratum
B. Panahbehagh and R. Bruggemann
Table 6 Example of POC and POR for a set with 5 units with R = 2. According to Poset, there is
an order between all the units except b that is incomparable with c, d and e. For these units there
are 4 LEs which leads to an unequal size post-stratified initial sample. For these units, 4-th stratum
will be empty and instead we will have two units in 3-th stratum. In POC we use mean result of
the all LEs and for POR we just select one of them randomly
POC
POR
Set X
Y
LE1 LE2 LE3 LE4 Set MH Stratum LE3 Set Stratum
e
8
7
e
e
e
b
e
4.75 5
e
e
5
d
4
6
d
d
b
e
d
3.50 3
b
d
3
c
3
5
c
b
d
d
c
2.25 2
d
c
2
b
1
9
b
c
c
c
b
3.50 3
c
b
4
a
0
0
a
a
a
a
a
1.00 1
a
a
1
Table 7 General POC. Please note that Z {h}i = (X {h}i , X {h}i ) is a unit that has been fallen into
the h-th stratum after i − 1 units, according to its mean height MH in respective LEs
Ranks
1 (1-th Stratum) 2 (2-th Stratum) · · · m (m-th Stratum)
Z {1}1
Z {2}1
· · · Z {m}1
Z {1}2
Z {2}2
· · · Z {m}2
. . .
. . .
. . .
. . .
. . .
. . .
· · · Z {m}Km
Z {1}K 1
. . .
Z {2}K 2
to POC (see Table 7). Panahbehagh (2020) showed that we can estimate unbiasedly
the elements of µ by
µ POC = ( μ x.POC , μ y.POC ) =
m
h=1
W h ¯
Z {h} , W h =
K h
Km
, ¯
Z {h} =
1
d h
iis h
Z {h}i .
(2)
where Z {h}i = (X {h}i , Y {h}i ) is a unit that has been fallen into the h-th stratum after
i − 1 units, according to its MH in the respective LEs.
For POC, because K h ; h = 1, 2, . . . , m are random, with unknown distribution,
mean and variance it would be complicated to calculate the variance of µ POC .
Then it is better to present a design with equal size post stratified-initial sample.
For this purpose, POR is presented. In POR, just one of the LEs would be selected
to decide about putting the units in the strata. In example of Table 6, assume LE3
is selected, then we will put the units of the set in the strata according to ranks of
them in LE3. With repeating this procedure for K sets we will have a post-stratified
initial sample with equal size. Now with showing the sample unit in h-th stratum