On the use of ranked set sampling for estimating super-population total: Gamma population model

Document Type : Article

Authors

Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan

Abstract

Utilization of superpopulation models for estimation of population parameters is an advantageous
practice, when it is easy to recognize the relationship between the study variable with one or more
auxiliary variables. This article is concerned with estimation of finite population total under a new ranked set sampling approach, ranked set sampling without replacement (RSSWOR), using so called gamma population model (GPM). Behavior of the proposed estimator, in term of relative efficiency, is studied for various choices of a constant γ via Monte Carle experiment. The provided simulation study shows the superiority of the proposed estimator over existing estimator under same model. The sampling procedure, especially, aids in collecting data from a continuous production process.

Keywords


References
1. Fuller, W.A. Simple estimators for the mean of
skewed populations", Technical Report, Iowa State
University, Dept. of Statistics (1970).
2. Royall, R. An old approach to nite population
sampling theory", Journal of American Statistical
Association, 63, pp. 1269{1279 (1969).
3. Royall, R.M. and Cumberland, W.G. The nitepopulation
linear regression estimator and estimators
of its variance an empirical study", Journal of the
American Statistical Association, 76(376), pp. 924{930
(1981).
4. Godambe, V.P. A uni ed theory of sampling from
nite populations", Journal of the Royal Statistical
Society: Series B (Methodological), 17(2), pp. 269{278
(1955).
5. Godambe, V. and Joshi, V. Admissibility and Bayes
estimation in sampling nite populations", The Annals
of Mathematical Statistics, 36(6), pp. 1707{1722
(1965).
6. Basu, D., An Essay on the Logical Foundations of
Survey Sampling Part i, in Foundations of Statistical
Inference, eds. Godambe and Sprott, Holt, Rinehart
and Winston of Canada, Toronto, pp. 203{233 (1971).
7. Smith, T.M.F. The foundations of survey sampling:
a review", Journal of the Royal Statistical Society:
Series A (General), 139(2), pp. 183{195 (1976).
8. Sarndal, C.E., Thomsen, I., Hoem, J.M., Lindley,
D.V., Barndor -Nielsen, O., and Dalenius, T. Designbased
and model-based inference in survey sampling
[with discussion and reply]", Scandinavian Journal of
Statistics, pp. 27{52 (1978).
9. Smith, T.M.F. On the validity of inferences from
non-random samples", Journal of the Royal Statistical
Society: Series A (General), 146(4), pp. 394{403
(1983).
10. Royall, R.M. The model based (prediction) approach
to nite population sampling theory", Lecture Notes-
Monograph Series, 17, pp. 225{240 (1992).
11. Sarndal, C.E., Swensson, B., and Wretman, J., Model
Assisted Survey Sampling, Springer Science & Business
Media (2003).
12. Royall, R. The linear least-squares prediction approach
to two-stage sampling", Journal of American
Statistical Association, 71, pp. 657{664 (1976).
13. Hansen, M.H., Madow, W.G., and Tepping, B.J.
An evaluation of model-dependent and probabilitysampling
inferences in sample surveys", Journal of the
American Statistical Association, 78(384), pp. 776{793
(1983).
14. Rao, J. Development in sample survey theory", The
Canadian Journal of Statistics, 25, pp. 1{21 (1996).
15. Brewer, K.R., Combined Survey Sampling Inference:
Weighing Basu's Elephants, Oxford University Press
(2002).
S. Ahmed and J. Shabbir/Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 465{476 475
16. Brewer, K. and Gregoire, T.G. Introduction to survey
sampling", Handbook of Statistics, 29, pp. 9{37 (2009).
17. Valliant, R. Model-based prediction of nite population
totals", Sample Surveys: Inference and Analysis,
29B, pp. 23{31 (2009).
18. Cheruiyot, R., Cheruiyot, T., and Jepchumba, L.
Estimation of population total using model-based
approach: A case of hiv/aids in nakuru central district,
kenya", International Journal of Scienti c and
Technology Research, 3(11), pp. 171{175 (2014).
19. Podlaski, R. and Roesch, F.A. Modelling diameter
distributions of two-cohort forest stands with various
proportions of dominant species: a two-component
mixture model approach", Mathematical Biosciences,
249, pp. 60{74 (2014).
20. Bohning, D. Ratio plot and ratio regression with
applications to social and medical sciences", Statistical
Science, 31(2), pp. 205{218 (2016).
21. Ogundimu, E.O., Altman, D.G., and Collins, G.S.
Adequate sample size for developing prediction models
is not simply related to events per variable",
Journal of Clinical Epidemiology, 76, pp. 175{182
(2016).
22. Kumar, S., Sisodia, B.V.S., Singh, D., and Basak, P.
Calibration approach based estimation of nite population
total in survey sampling under super population
model when study variable and auxiliary variable are
inversely related", Journal of Reliability and Statistical
Studies, 10(2), pp. 83{93 (2017).
23. Lovasi, G.S., Fink, D.S., Mooney, S.J., and Link,
B.G. Model-based and design-based inference goals
frame how to account for neighborhood clustering in
studies of health in overlapping context types", SSMPopulation
Health, 3, pp. 600{608 (2017).
24. Li, J. Assessing the accuracy of predictive models for
numerical data: Not r nor r2, why not? then what?",
PloS One, 12(8), e0183250 (2017).
25. McIntyre, G. A method for unbiased selective sampling
using ranked sets", Crop and Pasture Science, 3,
pp. 385{390 (1952).
26. Dell, T. and Clutter, J. Ranked set sampling theory
with order statistics background", Biometrica, 28, pp.
545{555 (1972).
27. Patil, G., Sinha, A., and Taillie, C. Finite population
corrections for ranked set sampling", Annals of the
Institute of Statistical Mathematics, 47(4), pp. 621{
636 (1995).
28. Muttlak, H. Median ranked set sampling", Journal of
Applied Statistical Sciences, 6(4), pp. 577{586 (1997).
29. Al-Saleh, M.F. and Al-Omari, A.I. Multistage ranked
set sampling", Journal of Statistical Planning and
Inference, 102(2), pp. 273{286 (2002).
30. Mahdizadeh, M. and Zamanzade, E. Ecient body
fat estimation using multistage pair ranked set sampling",
Statistical Methods in Medical Research, SAGE
Publications Sage UK: London, England (2018).
31. Samawi, H.M. and Muttlak, H.A. Estimation of ratio
using rank set sampling", Biometrical Journal, 38, pp.
753{764 (1996).
32. Ohyama, T.D.J. and Yanagawa, T. Estimating population
characteristics by incorporating prior values
in strati ed random sampling/ranked set sampling",
Journal of Statistical Planning and Inference, 138, pp.
4021{4032 (1999).
33. Bouza, C. Ranked set subsampling the non-response
strata for estimating the di erence of means", Biometrical
Journal, 1, pp. 203{243 (2002).
34. Al-Omari, A. and Jaber, K. Percentile double ranked
set sampling", Journal of Mathematics and Statistics,
44, pp. 903{915 (2008).
35. Haq, A., Brown, J., Moltchanova, E., and Al-Omari,
A.I. Mixed ranked set sampling design", Journal of
Applied Statistics, 41(10), pp. 2141{2156 (2014).
36. Salehi, M. and Jafari, A. Estimation of stress-strength
reliability using record ranked ret sampling scheme
from the exponential distribution", Filomat, 29(5), pp.
1149{1162 (2015).
37. Ahmed, S. and Shabbir, J. Extreme-cum-median
ranked set sampling", Brazilian Journal of Probability
and Statistics, 33(1), pp. 24{38 (2019).
38. Priya, R. and Thomas, P.Y. An application of ranked
set sampling when observations from several distributions
are to be included in the sample", Communications
in Statistics-Theory and Methods, 45(23), pp.
7040{7052 (2016).
39. Mahdizadeh, M. and Zamanzade, E. Ecient body
fat estimation using multistage pair ranked set sampling",
Statistical Methods in Medical Research, 28(1),
pp. 223{234 (2019).
40. Dumbgen, L. and Zamanzade, E. Inference on a
distribution function from ranked set samples" Annals
of the Institute of Statistical Mathematics, 72(1), pp.
157{185 (2020).
41. Aitchison, J. and Dunsmore, I.R., Statistical Prediction
Analysis, Cambridge, MA: Cambridge University
Press (1975).
42. Bain, L.J., Statistical Analysis of Reliability and Life
Testing Model, New York, NY: Marcel Dekker (1978).
43. Sinha, S.K. On the prediction limits for Rayleigh life
distribution", Calcutta Statistical Association Bulletin,
39, pp. 105{109 (1990).
44. Raqab, M.Z. Modi ed maximum likelihood predictors
of future order statistics from normal samples",
Computational Statistics and Data Analysis, 25, pp.
91{106 (1997).
476 S. Ahmed and J. Shabbir/Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 465{476
45. Raqab, M.Z. and Madi, M.T. Bayesian prediction of
the total time on test using doubly censored Rayleigh
data", Journal of Statistical Computational and Simulation,
72, pp. 781{789 (2002).
46. Chambers, R. and Clark, R., An Introduction to
Model-Based Survey Sampling with Applications, OUP
Oxford, 37 (2012).