Introduction into Sampling Theory, Applying Partial Order Concepts
141
V ( μ VSR ) −→ V ( μ RSS ) where μ RSS is the conventional estimator in the conventional
RSS (for more details about definition of μ RSS and its variance see Panahbehagh et al.
2018). As we can see, VSR is a kind of RSS that is executable with any K larger
than d. For example if we set m = 5, to have a ranked sample of size n • = d × m =
3 × 5 = 15, we can implement a VSR with selecting Km = 4 × 5 = 20 units while
for a conventional RSS we need d × m 2 = 3 × 25 = 75 units. Then VSR is a cost
efficient, easy to implement version of RSS. Panahbehagh et al. (2018) also showed
that if we define the efficiency based on high precision and low cost simultaneously,
VSR can be more efficient than RSS and LTR. In next section we extend such idea
to multivariate variables.
3 Multivariate Sampling Based on Ranks of Data
When we have just one variable, it is easy to rank and sort the sample units. For
example if we have two persons, it is straightforward to sort them based on their
heights or their weights. But it would be complicated or even impossible if we
want to sort them based on the two variables, height and weight simultaneously
(see Fig. 2).
In this section we will make a connection between sampling theory and partial
order set theory. For this purpose first we discuss few version of multivariate RSS,
involving multiple variable VSR.
3.1 A Multivariate Ranked Set Sampling
In the case of multivariate variables (Patil et al. 1994) considered one of the
variables as the main variable and sorted the units based this main variable. With
this approach, the strategy is efficient for estimating the mean of the main variable
and efficiency for the other variables depends on their correlations with the main
one. To consider all the variables (say R) simultaneously, often, however, too many
initial sample are taken, and then running a R-layer procedure to consider each
variable in a layer such that at the last we have all combinations of all the ranks for
all the variables together (for more details about such strategies see Al-Saleh and
Zheng 2002; Chen and Shen 2003; Arnold et al. 2009). For example with R = 2
and m = 5, we need 5 4 = 625 initial units to perform such designs that indicates
inefficiency of them with considering cost.
After presenting a multivariate version of VSR based on the method of Patil
et al. (1994), in the next subsection, based on partial order set theory, a simple and
cost efficient version of multivariate RSS introduced by Panahbehagh (2020) will
be presented.
141
V ( μ VSR ) −→ V ( μ RSS ) where μ RSS is the conventional estimator in the conventional
RSS (for more details about definition of μ RSS and its variance see Panahbehagh et al.
2018). As we can see, VSR is a kind of RSS that is executable with any K larger
than d. For example if we set m = 5, to have a ranked sample of size n • = d × m =
3 × 5 = 15, we can implement a VSR with selecting Km = 4 × 5 = 20 units while
for a conventional RSS we need d × m 2 = 3 × 25 = 75 units. Then VSR is a cost
efficient, easy to implement version of RSS. Panahbehagh et al. (2018) also showed
that if we define the efficiency based on high precision and low cost simultaneously,
VSR can be more efficient than RSS and LTR. In next section we extend such idea
to multivariate variables.
3 Multivariate Sampling Based on Ranks of Data
When we have just one variable, it is easy to rank and sort the sample units. For
example if we have two persons, it is straightforward to sort them based on their
heights or their weights. But it would be complicated or even impossible if we
want to sort them based on the two variables, height and weight simultaneously
(see Fig. 2).
In this section we will make a connection between sampling theory and partial
order set theory. For this purpose first we discuss few version of multivariate RSS,
involving multiple variable VSR.
3.1 A Multivariate Ranked Set Sampling
In the case of multivariate variables (Patil et al. 1994) considered one of the
variables as the main variable and sorted the units based this main variable. With
this approach, the strategy is efficient for estimating the mean of the main variable
and efficiency for the other variables depends on their correlations with the main
one. To consider all the variables (say R) simultaneously, often, however, too many
initial sample are taken, and then running a R-layer procedure to consider each
variable in a layer such that at the last we have all combinations of all the ranks for
all the variables together (for more details about such strategies see Al-Saleh and
Zheng 2002; Chen and Shen 2003; Arnold et al. 2009). For example with R = 2
and m = 5, we need 5 4 = 625 initial units to perform such designs that indicates
inefficiency of them with considering cost.
After presenting a multivariate version of VSR based on the method of Patil
et al. (1994), in the next subsection, based on partial order set theory, a simple and
cost efficient version of multivariate RSS introduced by Panahbehagh (2020) will
be presented.