Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response

Ongoma Jackson; Alilah David Anekeya; Okuto Erick

doi:doi:10.11648/j.ijsd.20210701.13

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Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response

Ongoma Jackson, Alilah David Anekeya, Okuto Erick

Published in International Journal of Statistical Distributions and Applications (Volume 7, Issue 1)

Received: 3 November 2020 Accepted: 1 December 2020 Published: 12 March 2021

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Abstract

Sample survey provides reliable current statistics for large areas or sub-population (domains) with large sample sizes. There is a growing demand for reliable small area statistics, however, the sample sizes are too small to provide direct (or area specific) estimators with acceptable and reliable accuracy. This study gives theoretical description of the estimation of small area mean by use of stratified sampling with a linear cost function in the presence of non-response. The estimation of small area mean is proposed using auxiliary information in which the study and auxiliary variable suffers from non-response during sampling. Optimal sample sizes have been obtained by minimizing the cost of survey for specific precision within a given cost using lagrangian function multiplier lambda and Partial Differential Equations (PDEs). Results demonstrate that as the values of the respondent sample increases sample units that supply information to study and auxiliary variable tends to small area population size, the non-response sample unit tends to sample units that supply the information as the sampling rate tends to one. From theoretic analysis it is practical that the Mean Square Error will decrease as the sub-sampling fraction and auxiliary characters increase. As the sub-sampling fraction increases and the value of beta increases then the value of large sample size is minimized with a reduction of Lagrangian multiplier value which minimizes the cost function.

Published in	International Journal of Statistical Distributions and Applications (Volume 7, Issue 1)
DOI	10.11648/j.ijsd.20210701.13
Page(s)	13-24
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Stratified Sampling for Ratio Estimation, Small Area Mean, Auxiliary Variable, Linear Cost Function and Non-response

References

[1]	Abhishek. N (2013). An overview of Fay Herriot model with our package in small area.
[2]	Aditya K. (2014). Estimation of Domain Mean Using Two-Stage Sampling with Sub-Sampling Non-response. Journal of the Indian Society of Agricultural Statistics 68 (1) pp. 39-54.
[3]	Alilah D. A and Ouma C. O (2018). Domain Mean Estimation Using Double Sampling with Non-Linear Cost functions in the presence of Non-response. Science Journal of the Applied Mathematics and Statistics Vol 6, No 1, pp. 28-42.
[4]	Arnold G. S. Harslet and Noble N. (2002). Small Area Estimation via generalized linear model. Journal of official Statistics vol. 18 (1) pp 45-60.
[5]	Cochran W. G (1977). Sampling techniques, New York John Wiley and sons.
[6]	Chaundhary M. K, Kumar A. (2016). Estimation of Mean of Finite Population using Double sampling Scheme under Non-response, Journal of Mathematical Sciences 5 (2), pp 287-297.
[7]	Cherniyak O. I (2001). Optimal allocation in stratified sampling and double sampling with non-linear cost function, Journal of Mathematical Sciences 103, 4 pp. 525-528.
[8]	Hansen M. H and Hurwitz W. N (1946). The problem of non-response in sampling surveys, Journal of American Statistical Association.
[9]	Harville D. A (1977). Maximum Likelihood approaches to variance component estimation and related problems. Journal of American Statistical ass. 72 (320-340).
[10]	Henderson, C. R. (1975). Best linear unbiased estimation and prediction under selection model. Biometrics. 31 (423-447).
[11]	Lohr, S. L. (2010) Sampling Design and Analysis. Boston, MA 02210, USA: Brooks/Col, Cengage Learning.
[12]	Okafor F. C (2001) Treatment of non response in successive sampling, Statistica, 61 (2) pp. 195-204.
[13]	Rahman (2008). A review of small area estimation problems and methodological developments. Discussion paper NATSEM-University of Canberra.
[14]	Saini M. and Kumar A. (2015). Method of optimum allocation for multivariate stratified two stage sampling design using double sampling. Journal of probability and statistic forum, 8 pp 19-23.
[15]	Wanjoya A. K, Torelli N and Datta G. (2012). Small Area Estimation JAGST Vol 14 (1).

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APA Style

Ongoma Jackson, Alilah David Anekeya, Okuto Erick. (2021). Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. International Journal of Statistical Distributions and Applications, 7(1), 13-24. https://doi.org/10.11648/j.ijsd.20210701.13

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ACS Style

Ongoma Jackson; Alilah David Anekeya; Okuto Erick. Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. Int. J. Stat. Distrib. Appl. 2021, 7(1), 13-24. doi: 10.11648/j.ijsd.20210701.13

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AMA Style

Ongoma Jackson, Alilah David Anekeya, Okuto Erick. Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. Int J Stat Distrib Appl. 2021;7(1):13-24. doi: 10.11648/j.ijsd.20210701.13

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@article{10.11648/j.ijsd.20210701.13,
  author = {Ongoma Jackson and Alilah David Anekeya and Okuto Erick},
  title = {Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {7},
  number = {1},
  pages = {13-24},
  doi = {10.11648/j.ijsd.20210701.13},
  url = {https://doi.org/10.11648/j.ijsd.20210701.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20210701.13},
  abstract = {Sample survey provides reliable current statistics for large areas or sub-population (domains) with large sample sizes. There is a growing demand for reliable small area statistics, however, the sample sizes are too small to provide direct (or area specific) estimators with acceptable and reliable accuracy. This study gives theoretical description of the estimation of small area mean by use of stratified sampling with a linear cost function in the presence of non-response. The estimation of small area mean is proposed using auxiliary information in which the study and auxiliary variable suffers from non-response during sampling. Optimal sample sizes have been obtained by minimizing the cost of survey for specific precision within a given cost using lagrangian function multiplier lambda and Partial Differential Equations (PDEs). Results demonstrate that as the values of the respondent sample increases sample units that supply information to study and auxiliary variable tends to small area population size, the non-response sample unit tends to sample units that supply the information as the sampling rate tends to one. From theoretic analysis it is practical that the Mean Square Error will decrease as the sub-sampling fraction and auxiliary characters increase. As the sub-sampling fraction increases and the value of beta increases then the value of large sample size is minimized with a reduction of Lagrangian multiplier value which minimizes the cost function.},
 year = {2021}
}

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TY  - JOUR
T1  - Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response
AU  - Ongoma Jackson
AU  - Alilah David Anekeya
AU  - Okuto Erick
Y1  - 2021/03/12
PY  - 2021
N1  - https://doi.org/10.11648/j.ijsd.20210701.13
DO  - 10.11648/j.ijsd.20210701.13
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 13
EP  - 24
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20210701.13
AB  - Sample survey provides reliable current statistics for large areas or sub-population (domains) with large sample sizes. There is a growing demand for reliable small area statistics, however, the sample sizes are too small to provide direct (or area specific) estimators with acceptable and reliable accuracy. This study gives theoretical description of the estimation of small area mean by use of stratified sampling with a linear cost function in the presence of non-response. The estimation of small area mean is proposed using auxiliary information in which the study and auxiliary variable suffers from non-response during sampling. Optimal sample sizes have been obtained by minimizing the cost of survey for specific precision within a given cost using lagrangian function multiplier lambda and Partial Differential Equations (PDEs). Results demonstrate that as the values of the respondent sample increases sample units that supply information to study and auxiliary variable tends to small area population size, the non-response sample unit tends to sample units that supply the information as the sampling rate tends to one. From theoretic analysis it is practical that the Mean Square Error will decrease as the sub-sampling fraction and auxiliary characters increase. As the sub-sampling fraction increases and the value of beta increases then the value of large sample size is minimized with a reduction of Lagrangian multiplier value which minimizes the cost function.
VL  - 7
IS  - 1
ER  -

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Author Information

Ongoma Jackson

Department of Mathematics Masinde Muliro University of Science and Technology, Kakamega-Nairobi, Kenya
Alilah David Anekeya

Department of Mathematics Masinde Muliro University of Science and Technology, Kakamega-Nairobi, Kenya
Okuto Erick

Departments of Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Siaya-Nairobi, Kenya

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APA Style

Ongoma Jackson, Alilah David Anekeya, Okuto Erick. (2021). Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. International Journal of Statistical Distributions and Applications, 7(1), 13-24. https://doi.org/10.11648/j.ijsd.20210701.13

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ACS Style

Ongoma Jackson; Alilah David Anekeya; Okuto Erick. Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. Int. J. Stat. Distrib. Appl. 2021, 7(1), 13-24. doi: 10.11648/j.ijsd.20210701.13

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AMA Style

Ongoma Jackson, Alilah David Anekeya, Okuto Erick. Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response. Int J Stat Distrib Appl. 2021;7(1):13-24. doi: 10.11648/j.ijsd.20210701.13

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@article{10.11648/j.ijsd.20210701.13,
  author = {Ongoma Jackson and Alilah David Anekeya and Okuto Erick},
  title = {Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {7},
  number = {1},
  pages = {13-24},
  doi = {10.11648/j.ijsd.20210701.13},
  url = {https://doi.org/10.11648/j.ijsd.20210701.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20210701.13},
  abstract = {Sample survey provides reliable current statistics for large areas or sub-population (domains) with large sample sizes. There is a growing demand for reliable small area statistics, however, the sample sizes are too small to provide direct (or area specific) estimators with acceptable and reliable accuracy. This study gives theoretical description of the estimation of small area mean by use of stratified sampling with a linear cost function in the presence of non-response. The estimation of small area mean is proposed using auxiliary information in which the study and auxiliary variable suffers from non-response during sampling. Optimal sample sizes have been obtained by minimizing the cost of survey for specific precision within a given cost using lagrangian function multiplier lambda and Partial Differential Equations (PDEs). Results demonstrate that as the values of the respondent sample increases sample units that supply information to study and auxiliary variable tends to small area population size, the non-response sample unit tends to sample units that supply the information as the sampling rate tends to one. From theoretic analysis it is practical that the Mean Square Error will decrease as the sub-sampling fraction and auxiliary characters increase. As the sub-sampling fraction increases and the value of beta increases then the value of large sample size is minimized with a reduction of Lagrangian multiplier value which minimizes the cost function.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Optimal Allocation in Small Area Mean Estimation Using Stratified Sampling in the Presence of Non-Response
AU  - Ongoma Jackson
AU  - Alilah David Anekeya
AU  - Okuto Erick
Y1  - 2021/03/12
PY  - 2021
N1  - https://doi.org/10.11648/j.ijsd.20210701.13
DO  - 10.11648/j.ijsd.20210701.13
T2  - International Journal of Statistical Distributions and Applications
JF  - International Journal of Statistical Distributions and Applications
JO  - International Journal of Statistical Distributions and Applications
SP  - 13
EP  - 24
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20210701.13
AB  - Sample survey provides reliable current statistics for large areas or sub-population (domains) with large sample sizes. There is a growing demand for reliable small area statistics, however, the sample sizes are too small to provide direct (or area specific) estimators with acceptable and reliable accuracy. This study gives theoretical description of the estimation of small area mean by use of stratified sampling with a linear cost function in the presence of non-response. The estimation of small area mean is proposed using auxiliary information in which the study and auxiliary variable suffers from non-response during sampling. Optimal sample sizes have been obtained by minimizing the cost of survey for specific precision within a given cost using lagrangian function multiplier lambda and Partial Differential Equations (PDEs). Results demonstrate that as the values of the respondent sample increases sample units that supply information to study and auxiliary variable tends to small area population size, the non-response sample unit tends to sample units that supply the information as the sampling rate tends to one. From theoretic analysis it is practical that the Mean Square Error will decrease as the sub-sampling fraction and auxiliary characters increase. As the sub-sampling fraction increases and the value of beta increases then the value of large sample size is minimized with a reduction of Lagrangian multiplier value which minimizes the cost function.
VL  - 7
IS  - 1
ER  -

Copy | Download