Published

2014-01-01

Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design

Algoritmos para calcular probabilidades exactas de inclusión para un diseño de muestreo no rechazable pi*pt

DOI:

https://doi.org/10.15446/rce.v37n1.44362

Keywords:

AP sampling design, Inclusion probabilities, Poisson sampling (en)
AP diseño de muestra, probabilidades de inclusión, esquema de Poisson (es)

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Authors

  • Yuxia Xue Inner Mongolia University of Technology
  • Zaizai Yan Inner Mongolia University of Technology

AP-design, an efficient non-rejective implementation of the ps sampling design, was proposed in the literature as an alternative Poisson sampling scheme. In this paper, we have updated inclusion probabilities formulas in the AP sampling design. The formulas of these inclusion probabilities have been greatly simplified. The proposed results show that the AP design and the algorithms to calculate inclusion probabilities are simple and effective, and the design is possible to be used in practice. Three real examples have also been included to illustrate the performance of these designs.

Una implementación del diseño de muestreo pt, que no es de rechazo, ha sido recientemente propuesta como alternativa al esquema de Poisson. En este trabajo, hemos adaptado las formulas de probabilidades de inclusión en el diseño de muestreo Poisson alternativo (AP por sus siglas en inglés). Estas fórmulas han sido significativamente simplificadas. Los resultados propuestos muestran que el diseño AP y los algoritmos para calcular las probabilidades de inclusión son simples y efectivos, y que el diseño se puede usar en la práctica. Se incluyen tres ejemplos reales para ilustrar el desempeño de la propuesta.

https://doi.org/10.15446/rce.v37n1.44362

Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate πpsSampling Design

Algoritmos para calcular probabilidades exactas de inclusión para un diseño de muestreo no rechazable πpt

ZAIZAI YAN1, YUXIA XUE2

1Inner Mongolia University of Technology, Science College, Hohhot, P. R. China. Professor. Email: zz.yan@163.com
2Inner Mongolia University of Technology, Science College, Hohhot, P. R. China. Postgraduate student. Email: yuxiaxue_imut@163.com

Abstract

AP-design, an efficient non-rejective implementation of the πps sampling design, was proposed in the literature as an alternative Poisson sampling scheme. In this paper, we have updated inclusion probabilities formulas in the AP sampling design. The formulas of these inclusion probabilities have been greatly simplified. The proposed results show that the AP design and the algorithms to calculate inclusion probabilities are simple and effective, and the design is possible to be used in practice. Three real examples have also been included to illustrate the performance of these designs.

Key words: AP sampling design, Inclusion probabilities, Poisson sampling.

Resumen

Una implementación del diseño de muestreo πpt, que no es de rechazo, ha sido recientemente propuesta como alternativa al esquema de Poisson. En este trabajo, hemos adaptado las formulas de probabilidades de inclusión en el diseño de muestreo Poisson alternativo (AP por sus siglas en inglés). Estas fórmulas han sido significativamente simplificadas. Los resultados propuestos muestran que el diseño AP y los algoritmos para calcular las probabilidades de inclusión son simples y efectivos, y que el diseño se puede usar en la práctica. Se incluyen tres ejemplos reales para ilustrar el desempeño de la propuesta.

Palabras clave: AP diseño de muestra, probabilidades de inclusión, esquema de Poisson.

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References

1. A. Grafströ, (2009), 'Non-rejective implementations of the Sampford sampling design', Journal of Statistical Planning and Inference 139(6), 2111-2114.

2. A. R. Sen, (1953), 'On the estimate of variance in sampling with varying probabilities', Journal of the Indian Society of Agricultural Statistics 5, 119-127.

3. B. Rosén, (1997a), 'On sampling with probability proportional to size', Journal of Statistical Planning and Inference 62(2), 159-191.

4. B. Rosén, (1997b), 'Asymptotic theory for order sampling', Journal of Statistical Planning and Inference 62(2), 135-158.

5. C. Asok, & B. V. Sukhatme, (1976), 'On Sampford's procedure of unequal probability sampling without replacement', Journal of the American Statistical Association 71(365), 912-918.

6. Cem Kadilar, & Hulya Cingi, (2004), 'Ratio estimators in simple random sampling', Applied Mathematics and Computation 151(3), 893-902.

7. J. Durbin, (1967), 'Design of multi-stage surveys for the estimation of sampling errors', Journal of the Royal Statistical Society. Series C: Applied Statistics 16(2), 152-164.

8. J. Hájek, (1964), 'Asymptotic theory of rejective sampling with varying probabilities from a finite population', Annals of Mathematical Statistics 35(4), 1491-1523.

9. J. Hájek, (1981), Sampling from a Finite Population, Marcel Dekker, New York.

10. J. Olofsson, (2011), 'Algorithms to find exact inclusion probabilities for 2P\pips sampling designs', Lithuanian Mathematical Journal 51(3), 425-439.

11. K. R. W. Brewer, (1963), 'A model of systematic sampling with unequal probability', Australian and New Zealand Journal of Statistics 5(1), 5-13.

12. L. Bondesson, & A. Grafström, (2011), 'An extension of Sampford's method for unequal probability sampling', Scandinavian Journal of Statistics 38(2), 377-392.

13. L. Bondesson,, I. Traat, & A. Lundqvist, (2006), 'Pareto sampling versus conditional Poisson and Sampford sampling', Scandinavian Journal of Statistics 33(4), 699-720.

14. L. Bondesson, & L. D. Thorburn, (2008), 'A list sequential sampling method suitable for real-time sampling', Scandinavian Journal of Statistics 35(3), 466-483.

15. M. P. Singh, (1967), 'Multivariate product method of estimation for finite populations', Journal of the Indian Society of Agricultural Statistics 31, 375-378.

16. M. R. Sampford, (1967), 'On sampling without replacement with unequal probabilities of selection', Biometrika 54(3-4), 499-513.

17. N. Aires, (1999), 'Algorithms to find exact inclusion probabilities for conditional Poisson sampling and Pareto \pips', Methodology and Computing in Applied Probability Sampling Designs 1(4), 457-469.

18. T. Laitila, & J. Olofsson, (2011), 'A two-phase sampling scheme and \pips designs', Journal of Statistical Planning and Inference 141(5), 1646-1654.

19. Y Tillé, (2006), Sampling Algorithms, Springer, New York.

20. Y. Zaizai,, L. Miaomiao, & Y. Yalu, (2013), 'An efficient non-rejective implementation of the \pips sampling designs', Journal of Applied Statistics 40(4), 870-886.

[Recibido en octubre de 2013. Aceptado en abril de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv37n1a09,
    AUTHOR  = {Yan, Zaizai and Xue, Yuxia},
    TITLE   = {{Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate πpsSampling Design}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2014},
    volume  = {37},
    number  = {1},
    pages   = {127-140}
}

References

Aires, N. (1999), ‘Algorithms to find exact inclusion probabilities for conditional Poisson sampling and Pareto pi-ps’, Methodology and Computing in Applied Probability Sampling Designs 1(4), 457–469.

Asok, C. & Sukhatme, B. V. (1976), ‘On sampford’s procedure of unequal probability sampling without replacement’, Journal of the American Statistical Association 71(365), 912–918.

Bondesson, L. & Grafström, A. (2011), ‘An extension of Sampford’s method for unequal probability sampling’, Scandinavian Journal of Statistics 38(2), 377– 392.

Bondesson, L. & Thorburn, L. D. (2008), ‘A list sequential sampling method suitable for real-time sampling’, Scandinavian Journal of Statistics 35(3), 466–483.

Bondesson, L., Traat, I. & Lundqvist, A. (2006), ‘Pareto sampling versus conditional Poisson and Sampford sampling’, Scandinavian Journal of Statistics 33(4), 699–720.

Brewer, K. R. W. (1963), ‘A model of systematic sampling with unequal probability’, Australian and New Zealand Journal of Statistics 5(1), 5–13.

Durbin, J. (1967), ‘Design of multi-stage surveys for the estimation of sampling errors’, Journal of the Royal Statistical Society. Series C: Applied Statistics 16(2), 152–164.

Grafströ, A. (2009), ‘Non-rejective implementations of the Sampford sampling design’, Journal of Statistical Planning and Inference 139(6), 2111–2114.

Hájek, J. (1964), ‘Asymptotic theory of rejective sampling with varying probabilities from a finite population’, Annals of Mathematical Statistics 35(4), 1491–1523.

Hájek, J. (1981), Sampling from a Finite Population, Marcel Dekker, New York.

Kadilar, C. & Cingi, H. (2004), ‘Ratio estimators in simple random sampling’, Applied Mathematics and Computation 151(3), 893–902.

Laitila, T. & Olofsson, J. (2011), ‘A two-phase sampling scheme and ps designs’, Journal of Statistical Planning and Inference 141(5), 1646–1654.

Olofsson, J. (2011), ‘Algorithms to find exact inclusion probabilities for 2p pi ps sampling designs’, Lithuanian Mathematical Journal 51(3), 425–439.

Rosén, B. (1997a), ‘Asymptotic theory for order sampling’, Journal of Statistical Planning and Inference 62(2), 135–158.

Rosén, B. (1997b), ‘On sampling with probability proportional to size’, Journal of Statistical Planning and Inference 62(2), 159–191.

Sampford, M. R. (1967), ‘On sampling without replacement with unequal probabilities of selection’, Biometrika 54(3-4), 499–513.

Sen, A. R. (1953), ‘On the estimate of variance in sampling with varying probabilities’, Journal of the Indian Society of Agricultural Statistics 5, 119–127.

Singh, M. P. (1967), ‘Multivariate product method of estimation for finite populations’, Journal of the Indian Society of Agricultural Statistics 31, 375–378.

Tillé, Y. (2006), Sampling Algorithms, Springer, New York.

Zaizai, Y., Miaomiao, L. & Yalu, Y. (2013), ‘An efficient non-rejective implementation of the ps sampling designs’, Journal of Applied Statistics 40(4), 870–886.

How to Cite

APA

Xue, Y. & Yan, Z. (2014). Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design. Revista Colombiana de Estadística, 37(1), 127–140. https://doi.org/10.15446/rce.v37n1.44362

ACM

[1]
Xue, Y. and Yan, Z. 2014. Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design. Revista Colombiana de Estadística. 37, 1 (Jan. 2014), 127–140. DOI:https://doi.org/10.15446/rce.v37n1.44362.

ACS

(1)
Xue, Y.; Yan, Z. Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design. Rev. colomb. estad. 2014, 37, 127-140.

ABNT

XUE, Y.; YAN, Z. Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design. Revista Colombiana de Estadística, [S. l.], v. 37, n. 1, p. 127–140, 2014. DOI: 10.15446/rce.v37n1.44362. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44362. Acesso em: 7 nov. 2025.

Chicago

Xue, Yuxia, and Zaizai Yan. 2014. “Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design”. Revista Colombiana De Estadística 37 (1):127-40. https://doi.org/10.15446/rce.v37n1.44362.

Harvard

Xue, Y. and Yan, Z. (2014) “Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design”, Revista Colombiana de Estadística, 37(1), pp. 127–140. doi: 10.15446/rce.v37n1.44362.

IEEE

[1]
Y. Xue and Z. Yan, “Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design”, Rev. colomb. estad., vol. 37, no. 1, pp. 127–140, Jan. 2014.

MLA

Xue, Y., and Z. Yan. “Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design”. Revista Colombiana de Estadística, vol. 37, no. 1, Jan. 2014, pp. 127-40, doi:10.15446/rce.v37n1.44362.

Turabian

Xue, Yuxia, and Zaizai Yan. “Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design”. Revista Colombiana de Estadística 37, no. 1 (January 1, 2014): 127–140. Accessed November 7, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44362.

Vancouver

1.
Xue Y, Yan Z. Algorithms to Calculate Exact Inclusion Probabilities for a Non-Rejective Approximate ps Sampling Design. Rev. colomb. estad. [Internet]. 2014 Jan. 1 [cited 2025 Nov. 7];37(1):127-40. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44362

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