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B. Panahbehagh and R. Bruggemann
one. Therefore it is an important task to develop still better approximations. An
alternative is to develop a theoretical framework to decide when which of the two
methods is to be preferred.
The set of linear extensions consists of LT elements (LE l , l = 1, . . . , LT ).
Assume that the variables used to describe the objects (or units) are of different
importance, then it is clear that linear extensions may have different proximities
to the variables (supposed there is a suitable concept of distances). Hence in a
general framework, the set of linear extensions should not be seen as a uniform
set. Especially if a variable induces a complete order (i.e. an order without ties)
then there must be one linear extension, which reproduces this order. This linear
extension may play a favorite role. Therefore in the further development of the
application of partial order sets in sampling theory, the potential non-uniformity of
linear extensions should be conceptually be built in. Linear extensions are images of
order-preserving maps, where the order relations found in a Poset are reproduced.
The underlying Poset derived from a data matrix reveals more symmetries than
expected from the data matrix alone, because of the ordinal interpretation of the
data. Hence the MH-values (see Table 6) may have several ties. It can be a useful
idea, to develop tie-breaking concepts, to guarantee that within the POC-concept the
objects belong as much as possible to different strata.
References
Al-Saleh, M., & Zheng, G. (2002). Estimation of bivariate characteristics using ranked set
sampling. The Australian & New Zealand Journal of Statistics, 44, 221–232.
Arnold, B. C., Castillo, E., & Sarabia, J. M. (2009). On multivariate order statistics. Application to
ranked set sampling. Computational Statistics and Data Analysis, 53(12), 4555–4569.
Bouza-Herrera, C., & Al-Omari, A. I. F. (2018). Ranked set sampling, 65 years improving the
accuracy in data gathering. London/San Diego: Elsevier/Academic.
Bruggemann, R., & Carlsen, L. (2011). An improved estimation of averaged ranks of partial orders.
MATCH Communications in Mathematical and in Computer Chemistry, 65, 383–414.
Bruggemann, R., Sorensen, P. B., Lerche, D., & Carlsen, L. (2004). Estimation of averaged ranks
by a local partial order model. Journal of Chemical Information and Computer Sciences, 44,
618–625.
Bubley, R., & Dyer, M. (1999). Faster random generation of linear extensions. Discrete Mathematics, 201, 81–88.
Chen, Z., & Shen, L. (2003). Two-layer ranked set sampling with concomitant variables. The
Journal of Statistical Planning and Inference, 115, 45–57.
Chen, Z., Bai, Z., & Sinha, B. (2004). Ranked set sampling: Theory and applications. Lecture
notes in statistics. New York: Springer.
Cochran, W. G. (1953). Sampling techniques. Oxford: Wiley.
Deville, J. C., & Sarndal, C. E. (1992). Calibration estimators in survey sampling. Journal of the
American Statistical Association, 87, 376–382.
Deng, L., & Chhikara, R. (1990). On the Ratio and Regression Estimation in Finite Population
Sampling. The American Statistician, 44(4), 282–284.
Deville, J. C., & Tille, Y. (2004). Efficient balanced sampling: The cube method. Biometrika, 91,
893–912.
B. Panahbehagh and R. Bruggemann
one. Therefore it is an important task to develop still better approximations. An
alternative is to develop a theoretical framework to decide when which of the two
methods is to be preferred.
The set of linear extensions consists of LT elements (LE l , l = 1, . . . , LT ).
Assume that the variables used to describe the objects (or units) are of different
importance, then it is clear that linear extensions may have different proximities
to the variables (supposed there is a suitable concept of distances). Hence in a
general framework, the set of linear extensions should not be seen as a uniform
set. Especially if a variable induces a complete order (i.e. an order without ties)
then there must be one linear extension, which reproduces this order. This linear
extension may play a favorite role. Therefore in the further development of the
application of partial order sets in sampling theory, the potential non-uniformity of
linear extensions should be conceptually be built in. Linear extensions are images of
order-preserving maps, where the order relations found in a Poset are reproduced.
The underlying Poset derived from a data matrix reveals more symmetries than
expected from the data matrix alone, because of the ordinal interpretation of the
data. Hence the MH-values (see Table 6) may have several ties. It can be a useful
idea, to develop tie-breaking concepts, to guarantee that within the POC-concept the
objects belong as much as possible to different strata.
References
Al-Saleh, M., & Zheng, G. (2002). Estimation of bivariate characteristics using ranked set
sampling. The Australian & New Zealand Journal of Statistics, 44, 221–232.
Arnold, B. C., Castillo, E., & Sarabia, J. M. (2009). On multivariate order statistics. Application to
ranked set sampling. Computational Statistics and Data Analysis, 53(12), 4555–4569.
Bouza-Herrera, C., & Al-Omari, A. I. F. (2018). Ranked set sampling, 65 years improving the
accuracy in data gathering. London/San Diego: Elsevier/Academic.
Bruggemann, R., & Carlsen, L. (2011). An improved estimation of averaged ranks of partial orders.
MATCH Communications in Mathematical and in Computer Chemistry, 65, 383–414.
Bruggemann, R., Sorensen, P. B., Lerche, D., & Carlsen, L. (2004). Estimation of averaged ranks
by a local partial order model. Journal of Chemical Information and Computer Sciences, 44,
618–625.
Bubley, R., & Dyer, M. (1999). Faster random generation of linear extensions. Discrete Mathematics, 201, 81–88.
Chen, Z., & Shen, L. (2003). Two-layer ranked set sampling with concomitant variables. The
Journal of Statistical Planning and Inference, 115, 45–57.
Chen, Z., Bai, Z., & Sinha, B. (2004). Ranked set sampling: Theory and applications. Lecture
notes in statistics. New York: Springer.
Cochran, W. G. (1953). Sampling techniques. Oxford: Wiley.
Deville, J. C., & Sarndal, C. E. (1992). Calibration estimators in survey sampling. Journal of the
American Statistical Association, 87, 376–382.
Deng, L., & Chhikara, R. (1990). On the Ratio and Regression Estimation in Finite Population
Sampling. The American Statistician, 44(4), 282–284.
Deville, J. C., & Tille, Y. (2004). Efficient balanced sampling: The cube method. Biometrika, 91,
893–912.