Published

2013-07-01

Censored Bimodal Symmetric-Asymmetric Alpha-Power Model

Modelo bimodal censurado simétrico-asimétrico alpha-potencia

Keywords:

AART, alpha-power model, bimodality, censorship, cumulative distribution, HIV-1 RNA, limit of detection, power-normal model. (en)
AART, bimodalidad, censura, distribución acumulada, HIV-1 RNA, límite de detección, modelo alfa potencia, modelo normal potencia (es)

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Authors

  • Hugo S. Salinas Universidad de Atacama
  • Guillermo Martínez-Flórez Universidad de Cordoba
  • Germán Moreno-Arenas Universidad Industrial de Santander
We introduce the censored bimodal symmetric-asymmetric alpha-power models to adjust censored data with bimodality and high levels of skewness and kurtosis. The moments corresponding are computed, the maximum likelihood estimation for the model parameters is considered and the observed information matrix is derived. We show the appropriateness of the proposed models through two applications with censored real data related to HIV-1 RNA measurement

Se introducen los modelos potencia alfa simétricos asimétricos bimodales censurados con el fin de ajustar datos censurados con bimodalidad y altos niveles de sesgo y curtosis. Los momentos correspondientes son calculados, se considera la estimación máximo verosímil para los parámetros del modelo y se deriva la matriz de información observada. Se muestra la utilidad de los modelos propuestos a través de dos aplicaciones con datos censurados reales
relacionados con la medición de HIV-1 RNA. 

Censored Bimodal Symmetric-Asymmetric Alpha-Power Model

Modelo bimodal censurado simétrico-asimétrico alpha-potencia

HUGO S. SALINAS1, GUILLERMO MARTÍNEZ-FLÓREZ2, GERMÁN MORENO-ARENAS3

1Universidad de Atacama, Facultad de Ingeniería, Departamento de Matemáticas, Copiapó, Chile. Associate professor. Email: hugo.salinas@uda.cl
2Universidad de Córdoba, Facultad de Ciencias, Departamento de Matemáticas y Estadística, Córdoba, Colombia. Professor. Email: gmartinez@correo.unicordoba.edu.co
3Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia. Professor. Email: gmorenoa@uis.edu.co

Abstract

We introduce the censored bimodal symmetric-asymmetric alpha-power models to adjust censored data with bimodality and high levels of skewness and kurtosis. The moments corresponding are computed, the maximum likelihood estimation for the model parameters is considered and the observed information matrix is derived. We show the appropriateness of the proposed models through two applications with censored real data related to HIV-1 RNA measurement.

Key words: AART, alpha-power model, bimodality, censorship, cumulative distribution, HIV-1 RNA, limit of detection, power-normal model.

Resumen

Se introducen los modelos potencia alfa simétricos asimétricos bimodales censurados con el fin de ajustar datos censurados con bimodalidad y altos niveles de sesgo y curtosis. Los momentos correspondientes son calculados, se considera la estimación máximo verosímil para los parámetros del modelo y se deriva la matriz de información observada. Se muestra la utilidad de los modelos propuestos a través de dos aplicaciones con datos censurados reales relacionados con la medición de HIV-1 RNA.

Palabras clave: AART, bimodalidad, censura, distribución acumulada, HIV-1 RNA, límite de detección, modelo alfa potencia, modelo normal potencia.

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References

1. Akaike, H. A. (1974), 'A new look at statistical model identification', IEEE Transaction on Automatic Control 19(6), 716-723.

2. Arellano-Valle, R., Castro, L., González-Farías, G. & Muñoz-Gajardo, K. (2012), 'Student-t censored regression model: Properties and inference', Statistical Methods and Applications 21(4), 453-473.

3. Arnold, B. C., Gómez, H. W. & Salinas, H. S. (2009), 'On multiple constraint skewed models', Statistics 43(3), 279-293.

4. Azzalini, A. (1985), 'A class of distributions which includes the normal ones', Scandinavian Journal of Statistics 12(2), 171-178.

5. Bolfarine, H., Gómez, H. W. & Rivas, L. I. (2011), 'The log-bimodal-skew-normal model. A geochemical application', Journal of Chemometrics 25(6), 329-332.

6. Bolfarine, H., Martínez-Flórez, G. & Salinas, H. S. (2012), 'Bimodal symmetric-asymmetric power-normal families', Communications in Statistics-Theory and Methods. DOI:10.1080/03610926.2013.765475.

7. Chu, H., Moulton, L. H., Mack, W. J., Passaro, D. J., Barroso, P. F. & Muñoz, A. (2005), 'Correlating two continuous variables subject to detection limits in the context of mixture distributions', Journal of the Royal Statistical Society. Series C (Applied Statistics) 54, 831-845.

8. Durrans, S. R. (1992), 'Distributions of fractional order statistics in hydrology', Water Resources Research 28(6), 1649-1655.

9. Elal-Olivero, D., Gómez, H. W. & Quintana, F. A. (2009), 'Bayesian modeling using a class of bimodal skew-elliptical distributions', Journal of Statistical Planning and Inference 139, 1484-1492.

10. Eugene, N., Lee, C. & Famoye, F. (2002), 'Beta-normal distribution and its applications', Communications in Statistics-Theory and Methods 31(4), 497-512.

11. Fernández, C. & Steel, M. F. J. (1998), 'On Bayesian modeling of fat tails and skewness', Journal of the American Statistical Association 93(441), 359-371.

12. Gómez, H. W., Elal-Olivero, D., Salinas, H. S. & Bolfarine, H. (2009), 'Bimodal extension based on the skew-normal distribution with application to pollen data', Environmetrics 22(1), 50-62.

13. Gómez, H. W., Venegas, O. & Bolfarine, H. (2007), 'Skew-symmetric distributions generated by the distribution function of the normal distribution', Environmetrics 18, 395-407.

14. Henze, N. (1986), 'A probabilistic representation of the skew-normal distribution', Scandinavian Journal of Statistics 13(4), 271-275.

15. Jones, M. C. (2004), 'Families of distributions arising from distributions of order statistics', TEST 13(1), 1-43.

16. Kim, H. J. (2005), 'On a class of two-piece skew-normal distribution', Statistic 39(6), 537-553.

17. Li, X., Chu, H., Gallant, J. E., Hoover, D. R., Mack, W. J., Chmiel, J. S. & Muñoz, A. (2006), 'Bimodal virological response to antiretroviral therapy for HIV infection: an application using a mixture model with left censoring', Journal of Epidemiology and Community Health 60(9), 811-818.

18. Martínez-Flórez, G., Bolfarine, H. & Gómez, H. W. (2013), 'The alpha-power tobit model', Communications in Statistics-Theory and Methods 42(4), 633-643.

19. Mudholkar, G. S. & Hutson, A. D. (2000), 'The epsilon-skew-normal distribution for analyzing near-normal data', Journal of Statistical Planning and Inference 83(2), 291-309.

20. Pewsey, A. (2000), 'Problems of inference for Azzalini's skew normal distribution', Journal of Applied Statistics 27(7), 859-870.

21. Schneider, M., Margolick, J., Jacobson, L., Reddy, S., Martinez-Maza, O. & Muñoz, A. (2012), 'Improved estimation of the distribution of suppressed plasma HIV-1 RNA in men receiving effective antiretroviral therapy', Journal of Acquired Immune Deficiency Syndromes 59(4), 389-392.

22. Schwarz, G. (1978), 'Estimating the dimension of a model', Annals of Statistics 6(2), 461-464.

23. Teck-Onn, L., Bakri, R., Morad, Z. & Hamid, M. A. (2002), 'Bimodality in blood glucose distribution', Diabetes Care 25(12), 2212-2217.

24. Tobin, J. (1958), 'Estimation of relationships for limited dependent variables', Econometrica (The Econometric Society) 26(1), 24-36.

[Recibido en enero de 2013. Aceptado en agosto de 2013]

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

@ARTICLE{RCEv36n2a07,
    AUTHOR  = {Salinas, Hugo S. and Martínez-Flórez, Guillermo and Moreno-Arenas, Germán},
    TITLE   = {{Censored Bimodal Symmetric-Asymmetric Alpha-Power Model}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {2},
    pages   = {285-301}
}

References

Akaike, H. A. (1974), ‘A new look at statistical model identification’, IEEE Transaction on Automatic Control 19(6), 716–723.

Arellano-Valle, R., Castro, L., González-Farías, G. & Muñoz-Gajardo, K. (2012), ‘Student-t censored regression model: Properties and inference’, Statistical Methods and Applications 21(4), 453–473.

Arnold, B. C., Gómez, H. W. & Salinas, H. S. (2009), ‘On multiple constraint skewed models’, Statistics 43(3), 279–293.

Azzalini, A. (1985), ‘A class of distributions which includes the normal ones’, Scandinavian Journal of Statistics 12(2), 171–178.

Bolfarine, H., Gómez, H. W. & Rivas, L. I. (2011), ‘The log-bimodal-skew-normal model. A geochemical application’, Journal of Chemometrics 25(6), 329–332.

Bolfarine, H., Martínez-Flórez, G. & Salinas, H. S. (2012), ‘Bimodal symmetricasymmetric power-normal families’, Communications in Statistics-Theory and Methods. DOI:10.1080/03610926.2013.765475.

Chu, H., Moulton, L. H., Mack, W. J., Passaro, D. J., Barroso, P. F. & Muñoz, A. (2005), ‘Correlating two continuous variables subject to detection limits in the context of mixture distributions’, Journal of the Royal Statistical Society. Series C (Applied Statistics) 54, 831–845.

Durrans, S. R. (1992), ‘Distributions of fractional order statistics in hydrology’, Water Resources Research 28(6), 1649–1655.

Elal-Olivero, D., Gómez, H. W. & Quintana, F. A. (2009), ‘Bayesian modeling using a class of bimodal skew-elliptical distributions’, Journal of Statistical Planning and Inference 139, 1484–1492.

Eugene, N., Lee, C. & Famoye, F. (2002), ‘Beta-normal distribution and its applications’, Communications in Statistics-Theory and Methods 31(4), 497–512.

Fernández, C. & Steel, M. F. J. (1998), ‘On Bayesian modeling of fat tails and skewness’, Journal of the American Statistical Association 93(441), 359–371.

Gómez, H. W., Elal-Olivero, D., Salinas, H. S. & Bolfarine, H. (2009), ‘Bimodal extension based on the skew-normal distribution with application to pollen data’, Environmetrics 22(1), 50–62.

Gómez, H. W., Venegas, O. & Bolfarine, H. (2007), ‘Skew-symmetric distributions generated by the distribution function of the normal distribution’, Environmetrics 18, 395–407.

Henze, N. (1986), ‘A probabilistic representation of the skew-normal distribution’, Scandinavian Journal of Statistics 13(4), 271–275.

Jones, M. C. (2004), ‘Families of distributions arising from distributions of order statistics’, TEST 13(1), 1–43.

Kim, H. J. (2005), ‘On a class of two-piece skew-normal distribution’, Statistic 39(6), 537–553.

Li, X., Chu, H., Gallant, J. E., Hoover, D. R., Mack, W. J., Chmiel, J. S. & Muñoz, A. (2006), ‘Bimodal virological response to antiretroviral therapy for HIV infection: an application using a mixture model with left censoring’, Journal of Epidemiology and Community Health 60(9), 811–818.

Martínez-Flórez, G., Bolfarine, H. & Gómez, H. W. (2013), ‘The alpha-power tobit model’, Communications in Statistics-Theory and Methods 42(4), 633–643.

Mudholkar, G. S. & Hutson, A. D. (2000), ‘The epsilon-skew-normal distribution for analyzing near-normal data’, Journal of Statistical Planning and Inference 83(2), 291–309.

Pewsey, A. (2000), ‘Problems of inference for Azzalini’s skew normal distribution’, Journal of Applied Statistics 27(7), 859–870.

Schneider, M., Margolick, J., Jacobson, L., Reddy, S., Martinez-Maza, O. & Muñoz, A. (2012), ‘Improved estimation of the distribution of suppressed plasma HIV-1 RNA in men receiving effective antiretroviral therapy’, Journal of Acquired Immune Deficiency Syndromes 59(4), 389–392.

Schwarz, G. (1978), ‘Estimating the dimension of a model’, Annals of Statistics 6(2), 461–464.

Teck-Onn, L., Bakri, R., Morad, Z. & Hamid, M. A. (2002), ‘Bimodality in blood glucose distribution’, Diabetes Care 25(12), 2212–2217.

Tobin, J. (1958), ‘Estimation of relationships for limited dependent variables’, Econometrica (The Econometric Society) 26(1), 24–36.

How to Cite

APA

Salinas, H. S., Martínez-Flórez, G. and Moreno-Arenas, G. (2013). Censored Bimodal Symmetric-Asymmetric Alpha-Power Model. Revista Colombiana de Estadística, 36(2), 285–301. https://revistas.unal.edu.co/index.php/estad/article/view/44349

ACM

[1]
Salinas, H.S., Martínez-Flórez, G. and Moreno-Arenas, G. 2013. Censored Bimodal Symmetric-Asymmetric Alpha-Power Model. Revista Colombiana de Estadística. 36, 2 (Jul. 2013), 285–301.

ACS

(1)
Salinas, H. S.; Martínez-Flórez, G.; Moreno-Arenas, G. Censored Bimodal Symmetric-Asymmetric Alpha-Power Model. Rev. colomb. estad. 2013, 36, 285-301.

ABNT

SALINAS, H. S.; MARTÍNEZ-FLÓREZ, G.; MORENO-ARENAS, G. Censored Bimodal Symmetric-Asymmetric Alpha-Power Model. Revista Colombiana de Estadística, [S. l.], v. 36, n. 2, p. 285–301, 2013. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44349. Acesso em: 28 mar. 2025.

Chicago

Salinas, Hugo S., Guillermo Martínez-Flórez, and Germán Moreno-Arenas. 2013. “Censored Bimodal Symmetric-Asymmetric Alpha-Power Model”. Revista Colombiana De Estadística 36 (2):285-301. https://revistas.unal.edu.co/index.php/estad/article/view/44349.

Harvard

Salinas, H. S., Martínez-Flórez, G. and Moreno-Arenas, G. (2013) “Censored Bimodal Symmetric-Asymmetric Alpha-Power Model”, Revista Colombiana de Estadística, 36(2), pp. 285–301. Available at: https://revistas.unal.edu.co/index.php/estad/article/view/44349 (Accessed: 28 March 2025).

IEEE

[1]
H. S. Salinas, G. Martínez-Flórez, and G. Moreno-Arenas, “Censored Bimodal Symmetric-Asymmetric Alpha-Power Model”, Rev. colomb. estad., vol. 36, no. 2, pp. 285–301, Jul. 2013.

MLA

Salinas, H. S., G. Martínez-Flórez, and G. Moreno-Arenas. “Censored Bimodal Symmetric-Asymmetric Alpha-Power Model”. Revista Colombiana de Estadística, vol. 36, no. 2, July 2013, pp. 285-01, https://revistas.unal.edu.co/index.php/estad/article/view/44349.

Turabian

Salinas, Hugo S., Guillermo Martínez-Flórez, and Germán Moreno-Arenas. “Censored Bimodal Symmetric-Asymmetric Alpha-Power Model”. Revista Colombiana de Estadística 36, no. 2 (July 1, 2013): 285–301. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44349.

Vancouver

1.
Salinas HS, Martínez-Flórez G, Moreno-Arenas G. Censored Bimodal Symmetric-Asymmetric Alpha-Power Model. Rev. colomb. estad. [Internet]. 2013 Jul. 1 [cited 2025 Mar. 28];36(2):285-301. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44349

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