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B. Panahbehagh and R. Bruggemann
• Representation: The exact definition of the representative sample has always
been the subject of various discussions in the statistical literature (for some
interesting discussions see Kruskal and Mosteller 1979a,b,c, 1980; Rao 2005;
Dumicic 2011; Tille and Wilhelm 2017). In fact, the definition of a representative
sample depends on our purpose. For example, if we are going to estimate the
population density of the respective variable, then a simple random sample (SRS)
would be a representative sample; if we are going to estimate the population
total, a sample proportional to size, which is not a miniature of the population
would be a representative sample (Tille and Wilhelm 2017) and if we are going
to know if any member in our population has a specified disease or not, then a
non-probability sample with many non-responses, containing just a few diseased
people, could be a representative sample.
In the way of searching for efficient strategies, based on the two principals,
randomization and representation, auxiliary variables have an important role in
the past, present, and probably future of sampling theory (Rao and Fuller 2017).
Some examples of the roles of auxiliary variables in the design stage are ranked set
sampling (McIntyre 1952; Chen et al. 2004; Bouza-Herrera and Al-Omari 2018),
judgment post-stratified sampling (MacEachern et al. 2004), balanced sampling
(Deville and Tille 2004) and for estimation stage are ratio and regression estimators
(Cochran 1953; Deng and Chhikara 1990), calibration estimator (Deville and
Sarndal 1992).
Among the strategies based on auxiliary variables, some of them like balanced
sampling, regression sampling, etc, need almost complete information about the
population of the auxiliary variables. For example, in balanced sampling, it is
assumed that we know the auxiliary variables for all the population units, before
starting the procedure of sampling, and in the regression estimator, it is assumed
that the population means of the auxiliary variables are known.
However some strategies just need partial information about the auxiliary
variables like ranked set sampling (RSS) and judgment post-stratified sampling.
In these designs just it is assumed that we know or even measure easily auxiliary
variables for the sampled units and based on such information a more representative
sample will be achievable.
Here we are going to discuss the later kind of strategies (that just need partial
information about the auxiliary variables) intending to reduce costs and enhance the
precision for multivariate variables. Then this chapter proceeds as follows; in Sect. 2
for using the information of ranks of data in the univariate case, we discuss RSS
design and an economic version of RSS introduced by Panahbehagh et al. (2018),
in Sect. 3 based on partial order set theory we discussed RSS for multivariate cases
based on a research of Panahbehagh (2020) and the chapter will be finished with a
conclusion in Sect. 4.
B. Panahbehagh and R. Bruggemann
• Representation: The exact definition of the representative sample has always
been the subject of various discussions in the statistical literature (for some
interesting discussions see Kruskal and Mosteller 1979a,b,c, 1980; Rao 2005;
Dumicic 2011; Tille and Wilhelm 2017). In fact, the definition of a representative
sample depends on our purpose. For example, if we are going to estimate the
population density of the respective variable, then a simple random sample (SRS)
would be a representative sample; if we are going to estimate the population
total, a sample proportional to size, which is not a miniature of the population
would be a representative sample (Tille and Wilhelm 2017) and if we are going
to know if any member in our population has a specified disease or not, then a
non-probability sample with many non-responses, containing just a few diseased
people, could be a representative sample.
In the way of searching for efficient strategies, based on the two principals,
randomization and representation, auxiliary variables have an important role in
the past, present, and probably future of sampling theory (Rao and Fuller 2017).
Some examples of the roles of auxiliary variables in the design stage are ranked set
sampling (McIntyre 1952; Chen et al. 2004; Bouza-Herrera and Al-Omari 2018),
judgment post-stratified sampling (MacEachern et al. 2004), balanced sampling
(Deville and Tille 2004) and for estimation stage are ratio and regression estimators
(Cochran 1953; Deng and Chhikara 1990), calibration estimator (Deville and
Sarndal 1992).
Among the strategies based on auxiliary variables, some of them like balanced
sampling, regression sampling, etc, need almost complete information about the
population of the auxiliary variables. For example, in balanced sampling, it is
assumed that we know the auxiliary variables for all the population units, before
starting the procedure of sampling, and in the regression estimator, it is assumed
that the population means of the auxiliary variables are known.
However some strategies just need partial information about the auxiliary
variables like ranked set sampling (RSS) and judgment post-stratified sampling.
In these designs just it is assumed that we know or even measure easily auxiliary
variables for the sampled units and based on such information a more representative
sample will be achievable.
Here we are going to discuss the later kind of strategies (that just need partial
information about the auxiliary variables) intending to reduce costs and enhance the
precision for multivariate variables. Then this chapter proceeds as follows; in Sect. 2
for using the information of ranks of data in the univariate case, we discuss RSS
design and an economic version of RSS introduced by Panahbehagh et al. (2018),
in Sect. 3 based on partial order set theory we discussed RSS for multivariate cases
based on a research of Panahbehagh (2020) and the chapter will be finished with a
conclusion in Sect. 4.