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Estimating Population Proportions by Means of Calibration Estimators
Estimación de proporciones poblacionales mediante estimadores de calibración
DOI:
https://doi.org/10.15446/rce.v38n1.48814Keywords:
Auxiliary Information, Calibration, Estimators, Finite Population, Sampling Design (en)Calibración, Diseño muestral, Estimadores, Información auxiliar, Población finita (es)
This paper considers the problem of estimating the population proportion of a categorical variable using the calibration framework. Different situations are explored according to the level of auxiliary information available and the theoretical properties are investigated. A new class of estimator based upon the proposed calibration estimators is also defined, and the optimal estimator in the class, in the sense of minimal variance, is derived. Finally, an estimator of the population proportion, under new calibration conditions, is defined. Simulation studies are considered to evaluate the performance of the proposed calibration estimators via the empirical relative bias and the empirical relative efficiency, and favourable results are achieved.
El artículo considera el problema de la estimación de la proporción poblacional de una variable categórica usando como marco de trabajo la calibración. Se exploran diferentes situaciones de acuerdo con la información auxiliar disponible y se investigan las propiedades teóricas.. Una nueva clase de estimadores basada en los estimadores de calibración propuestos también es definida y el estimador óptimo en la clase, en el sentido de varianza mínima, es obtenido. Finalmente, un estimador de la proporción poblacional, bajo nuevas condiciones de calibración es también propuesto. Estudios de simulación para evaluar el comportamiento de los estimadores calibrados propuestos a través del sesgo relativo empírico y de la eficiencia relativa empírica son incluidos, obteniéndose resultados satisfactorios.
https://doi.org/10.15446/rce.v38n1.48814
1University of Almería, Math Department, Almería, España. Professor. Email: spuertas@ual.es
2University of Granada, Department of Statistics and Operational Research, Granada, España. Professor. Email: arcos@ugr.es
3University of Almería, Math Department, Almería, España. Ph.D. Research Assistant. Email: hmartinez@ual.es
4Texas A&M University-Kingsville, Department of Mathematics, Kingsville Texas, United States. Associate Professor. Email: sarjinder.singh@tamuk.edu
This paper considers the problem of estimating the population proportion of a categorical variable using the calibration framework. Different situations are explored according to the level of auxiliary information available and the theoretical properties are investigated. A new class of estimator based upon the proposed calibration estimators is also defined, and the optimal estimator in the class, in the sense of minimal variance, is derived. Finally, an estimator of the population proportion, under new calibration conditions, is defined. Simulation studies are considered to evaluate the performance of the proposed calibration estimators via the empirical relative bias and the empirical relative efficiency, and favourable results are achieved.
Key words: Auxiliary Information, Calibration, Estimators, Finite Population, Sampling Design.
El artículo considera el problema de la estimación de la proporción poblacional de una variable categórica usando como marco de trabajo la calibración. Se exploran diferentes situaciones de acuerdo con la información auxiliar disponible y se investigan las propiedades teóricas. Una nueva clase de estimadores basada en los estimadores de calibración propuestos también es definida y el estimador óptimo en la clase, en el sentido de varianza mínima, es obtenido. Finalmente, un estimador de la proporción poblacional, bajo nuevas condiciones de calibración es también propuesto. Estudios de simulación para evaluar el comportamiento de los estimadores calibrados propuestos a trav\es del sesgo relativo empírico y de la eficiencia relativa empírica son incluidos, obteniéndose resultados satisfactorios.
Palabras clave: calibración, diseño muestral, estimadores, información auxiliar, población finita.
Texto completo disponible en PDF
References
1. Arnab, R., Shangodoyin, D. K. & Singh, S. (2010), 'Variance estimation of a generalized regression predictor', Journal of the Indian Society of Agricultural Statistics 64(2), 273-288.
2. Deville, Jean-Claude & S¤rndal, Carl-Erik (1992), 'Calibration estimators in survey sampling', Journal of the American Statistical Association 87(418), 376-382.
3. Duchesne, P. (2003), 'Estimation of a proportion with survey data', Journal of Statistics Education 11(3), 1-24.
4. Farrell, P. & Singh, S. (2005), 'Model-assisted higher-order calibration of estimators of variance', Australian & New Zealand Journal of Statistics 47(3), 375-383.
5. Harms, T. & Duchesne, P. (2006), 'On calibration estimation for quantiles', Survey Methodology 32, 37-52.
6. Rueda, M., Martínez, S., Martínez, H. & Arcos, A. (2007), 'Estimation of the distribution function with calibration methods', Journal of Statistical Planning and Inference 137(2), 435-448.
7. Rueda, M., Martínez-Puertas, S., Martínez-Puertas, H. & Arcos, A. (2007), 'Calibration methods for estimating quantiles', Metrika 66(3), 355-371.
8. Rueda, M., Muñoz, J. F., Arcos, A., Álvarez, E. & Martínez, S. (2011), 'Estimators and confidence intervals for the proportion using binary auxiliary information with applications to pharmaceutical studies', Journal of Biopharmaceutical Statistics 21(3), 526-54.
9. Singh, H. P., Singh, S. & Kozak, M. (2008), 'A family of estimators of finite-population distribution functions using auxiliary information', Acta Applicandae Mathematicae 104(2), 115-130.
10. Singh, S. (2001), 'Generalized calibration approach for estimating variance in survey sampling', Annals of the Institute of Statistical Mathematics 53(2), 404-417.
11. Singh, S. (2003), Advanced Sampling Theory with Applications: How Michael 'selected' Amy, Kluwer Academic Publishers.
12. Singh, S., Horn, S., Chowdhury, S. & Yu, F. (1999), 'Calibration of the estimator of variance', Australian and New Zealand Journal of Statistics 41, 199-212.
13. S¤rndal, C. (2007), 'The calibration approach in survey theory and practice', Survey Methodology 33(2), 99-119.
14. Wu, C. & Sitter, Y. R. (2001), 'A model-calibration approach to using complete auxiliary information from survey data', Journals - American Statistical Association 96, 185-193.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv38n1a14,
AUTHOR = {Martínez, Sergio and Arcos, Antonio and Martínez, Helena and Singh, Sarjinder},
TITLE = {{Estimating Population Proportions by Means of Calibration Estimators}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2015},
volume = {38},
number = {1},
pages = {267-293}
}
References
Arnab, R., Shangodoyin, D. K. & Singh, S. (2010), ‘Variance estimation of a generalized regression predictor’, Journal of the Indian Society of Agricultural Statistics 64(2), 273–288.
Deville, J.-C. & Särndal, C.-E. (1992), ‘Calibration estimators in survey sampling’, Journal of the American Statistical Association 87(418), 376–382.
Duchesne, P. (2003), ‘Estimation of a proportion with survey data’, Journal of Statistics Education 11(3), 1–24.
Farrell, P. & Singh, S. (2005), ‘Model-assisted higher-order calibration of estimators of variance’, Australian & New Zealand Journal of Statistics 47(3), 375–383.
Harms, T. & Duchesne, P. (2006), ‘On calibration estimation for quantiles’, Survey Methodology 32, 37–52.
Rueda, M., Martínez-Puertas, S., Martínez-Puertas, H. & Arcos, A. (2007), ‘Calibration methods for estimating quantiles’, Metrika 66(3), 355–371.
Rueda, M., Martínez, S., Martínez, H. & Arcos, A. (2007), ‘Estimation of the distribution function with calibration methods’, Journal of Statistical Planning and Inference 137(2), 435–448.
Rueda, M., Muñoz, J. F., Arcos, A., Álvarez, E. & Martínez, S. (2011), ‘Estimators and confidence intervals for the proportion using binary auxiliary information with applications to pharmaceutical studies’, Journal of Biopharmaceutical Statistics 21(3), 526–54.
Särndal, C. (2007), ‘The calibration approach in survey theory and practice’, Survey Methodology 33(2), 99–119.
Singh, H. P., Singh, S. & Kozak, M. (2008), ‘A family of estimators of finitepopulation distribution functions using auxiliary information’, Acta Applicandae Mathematicae 104(2), 115–130.
Singh, S. (2001), ‘Generalized calibration approach for estimating variance in survey sampling’, Annals of the Institute of Statistical Mathematics 53(2), 404–417.
Singh, S. (2003), Advanced Sampling Theory with Applications: How Michael ’selected’ Amy, Kluwer Academic Publishers.
Singh, S., Horn, S., Chowdhury, S. & Yu, F. (1999), ‘Calibration of the estimator of variance’, Australian and New Zealand Journal of Statistics 41, 199–212.
Wu, C. & Sitter, Y. R. (2001), ‘A model-calibration approach to using complete auxiliary information from survey data’, Journals - American Statistical Association 96, 185–193.
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