Introduction into Sampling Theory, Applying Partial Order Concepts
137
2 Univariate Sampling Based on Ranks of Data
Using ranks of the main variables, based on auxiliary variables, it is possible to take
a representative sample from the target population. Panahbehagh (2020) showed that
if we have complete information about the ranks of all the main variables we can
estimate the population total with a great efficiency relative to SRS. Knowing the
ranks of all the main or even auxiliary variables is very ambitious but it would be
reasonable to assume that if we take a sample, before measuring the main variable,
we can rank these sample units based on some easy to measure auxiliary variables. If
it is possible, then it is better first to take an initial large sample, rank them based on
some easy to measure auxiliary variable, and then select a good, more representative
sample and measure the final sample units exactly based on the main variable. This
idea goes back to McIntyre (1952). He introduced RSS for estimating a kind of
crops, without extending theory. Takahasi and Wakimoto (1968) extended the theory
of RSS and showed that this design is more efficient (more precise) than SRS.
2.1 Ranked Set Sampling
The idea of RSS is simple and beautiful. As a simple example assume we are going
to estimate the mean height of the students in a college based on a sample of size 3.
In SRS we select 3 students randomly and estimate the mean population based on the
sample mean. In RSS, because it is easy and inexpensive to rank the students based
on their heights, we first select 3 students, sort it, and select the shortest student as
the first sample unit. For the second sample unit, we select another SRS sample of
size 3, sort them and select the middle student and for the last sample unit we select
another SRS and select the tallest student as the final sample and then measure the
heights of these three students exactly and estimate the population mean based on
them (see Fig. 1a).
Many different kinds of research, theoretically and practically showed that with
considering precision, RSS is more efficient than SRS in many different problems
(for some complete reviews of RSS see Chen et al. 2004; Bouza-Herrera and AlOmari 2018). But yet many researchers are reluctant to use RSS for gathering
their sample because different versions of RSS produce non-iid (independent and
identical) samples and then conventional inferences that are extended based on iid
samples assumption are not applicable for RSS samples. That’s why MacEachern
et al. (2004) introduced judgment post stratified sampling (JPS).
The idea of JPS is using the information of ranks of data just in the estimation
stage and not the design stage. Then we can have an SRS (that is an iid sample)
but take advantage of the ranks of the observations. Following the example of RSS,
for estimating the mean height of the student, assume we are going to select a JPS
sample of size 3. We select 3 students by SRS as the main sample and for indicating
rank for each of the 3 observations, we select 2 students by SRS (as an auxiliary
137
2 Univariate Sampling Based on Ranks of Data
Using ranks of the main variables, based on auxiliary variables, it is possible to take
a representative sample from the target population. Panahbehagh (2020) showed that
if we have complete information about the ranks of all the main variables we can
estimate the population total with a great efficiency relative to SRS. Knowing the
ranks of all the main or even auxiliary variables is very ambitious but it would be
reasonable to assume that if we take a sample, before measuring the main variable,
we can rank these sample units based on some easy to measure auxiliary variables. If
it is possible, then it is better first to take an initial large sample, rank them based on
some easy to measure auxiliary variable, and then select a good, more representative
sample and measure the final sample units exactly based on the main variable. This
idea goes back to McIntyre (1952). He introduced RSS for estimating a kind of
crops, without extending theory. Takahasi and Wakimoto (1968) extended the theory
of RSS and showed that this design is more efficient (more precise) than SRS.
2.1 Ranked Set Sampling
The idea of RSS is simple and beautiful. As a simple example assume we are going
to estimate the mean height of the students in a college based on a sample of size 3.
In SRS we select 3 students randomly and estimate the mean population based on the
sample mean. In RSS, because it is easy and inexpensive to rank the students based
on their heights, we first select 3 students, sort it, and select the shortest student as
the first sample unit. For the second sample unit, we select another SRS sample of
size 3, sort them and select the middle student and for the last sample unit we select
another SRS and select the tallest student as the final sample and then measure the
heights of these three students exactly and estimate the population mean based on
them (see Fig. 1a).
Many different kinds of research, theoretically and practically showed that with
considering precision, RSS is more efficient than SRS in many different problems
(for some complete reviews of RSS see Chen et al. 2004; Bouza-Herrera and AlOmari 2018). But yet many researchers are reluctant to use RSS for gathering
their sample because different versions of RSS produce non-iid (independent and
identical) samples and then conventional inferences that are extended based on iid
samples assumption are not applicable for RSS samples. That’s why MacEachern
et al. (2004) introduced judgment post stratified sampling (JPS).
The idea of JPS is using the information of ranks of data just in the estimation
stage and not the design stage. Then we can have an SRS (that is an iid sample)
but take advantage of the ranks of the observations. Following the example of RSS,
for estimating the mean height of the student, assume we are going to select a JPS
sample of size 3. We select 3 students by SRS as the main sample and for indicating
rank for each of the 3 observations, we select 2 students by SRS (as an auxiliary