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Asymmetric Regression Models Bernoulli/Log Proportional-Hazard Distribution
Modelos de regresión asimétrico Bernoulli/distribución Log Hazard proporcional
DOI:
https://doi.org/10.15446/rce.v37n1.44365Keywords:
Censoring, Fisher information matrix, Maximum likelihood estimators, Proportional hazard. (en)censura, estimadores de máxima verosimilitud, hazard proporcional, matriz de información de Fisher (es)
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In this paper we introduce a kind of asymmetric distribution for nonnegative data called log-proportional hazard distribution (LPHF). This new distribution is used to study an asymmetrical regression model for data with limited responses (censored) through the mixture of a Bernoulli distribution with logit link and the LPHF distribution. Properties of the LPHF distribution are studied, maximum likelihood parameter estimation and information matrices are addressed. An illustration with real data shows that the model is a new alternative for studies with positive data censored.
En este artículo se introduce una forma de distribución asimétrica para datos no-negativos llamada distribución log hazard proporcional (LPHF). Esta nueva distribución es usada para estudiar un modelo de regresión asimétrico para datos con respuestas limitadas (censuradas) a través de mezclas de una distribución Bernoulli con función link logit y la distribución LPHF. Propiedades de la distribución LPHF son estudiadas, se abordan las estimaciones de máxima verosimilitud de los parámetros y las matrices de información. Se presenta una ilustración con datos reales, donde se muestra que el modelo propuesto es una nueva alternativa para estudios con datos positivos censurados.
https://doi.org/10.15446/rce.v37n1.44365
1Universidad de Córdoba, Departamento de Matemáticas y Estadística, Montería, Colombia. Professor. Email: gmartinez@correo.unicordoba.edu.co
2Instituto Tecnológico Metropolitano, Facultad de Ciencias Exactas y Aplicadas, Medellín, Colombia. Associate professor. Email: carlosbarrera@itm.edu.co
In this paper we introduce a kind of asymmetric distribution for non-negative data called log-proportional hazard distribution (LPHF). This new distribution is used to study an asymmetrical regression model for data with limited responses (censored) through the mixture of a Bernoulli distribution with logit link and the LPHF distribution. Properties of the LPHF distribution are studied, maximum likelihood parameter estimation and information matrices are addressed. An illustration with real data shows that the model is a new alternative for studies with positive data censored.
Key words: Censoring, Fisher information matrix, Maximum likelihoodestimators, Proportional hazard.
En este artículo se introduce una forma de distribución asimétrica para datos no-negativos llamada distribución log hazard proporcional (LPHF). Esta nueva distribución es usada para estudiar un modelo de regresión asimétrico para datos con respuestas limitadas (censuradas) a través de mezclas de una distribución Bernoulli con función link logit y la distribución LPHF. Propiedades de la distribución LPHF son estudiadas, se abordan las estimaciones de máxima verosimilitud de los parámetros y las matrices de información. Se presenta una ilustración con datos reales, donde se muestra que el modelo propuesto es una nueva alternativa para estudios con datos positivos censurados.
Palabras clave: censura, estimadores de máxima verosimilitud, hazardproporcional, matriz de información de Fisher.
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References
1. A. Azzalini, (1985), 'A class of distributions which includes the normal ones', Scandinavian Journal of Statistics 12, 171-178.
2. A. Azzalini,, T. dal Cappello, & S. Kotz, (2003), 'Log-skew-normal and log-skew-t distributions as models for family income data', Journal of Income Distribution 11, 12-20.
3. G. Martínez-Florez,, G. Moreno-Arenas, & S. Vergara-Cardozo, (2013), 'Properties and Inference for Proportional Hazard Models', Revista Colombiana de Estadística 36(1), 95-114.
4. G. Martínez-Florez,, H. Bolfarine, & H. W. Gómez, (2013), 'Asymmetric regression models with limited responses with an application to antibody response to vaccine', Biometrical Journal 55, 156-172..
5. G. Mateu-Figueras,, . Pawlosky-Glanh, & C. Barcelo-Vidal, (2004), The natural law in geochemistry: Lognormal or log skew-normal?, '32th International Geological Congress', International Union of Soil Sciences, Florence, Italy, p. 1849-1858.
6. G. Mateu-Figueras, & Pawlosky-Glanh, (2003), Una alternativa a la distribución log-normal, 'Actas del XXVII Congreso Nacional de Estadística e Investigación Operativa (SEIO)', Sociedade de Estadítica e Investigación Operativa, España, p. 1849-1858.
7. H. Akaike, (1974), 'A new look at statistical model identification', IEEE Transaction on Automatic Control AU-19, 716-722.
8. H. Chai, & K. Bailey, (2008), 'Use of log-normal distribution in analysis of continuous data with a discrete component at zero', Statistics in Medicine 27, 3643-3655.
9. J. Cragg, (1971), 'Some statistical models for limited dependent variables with application to the demand for durable goods', Econometrica 39, 829-844.
10. J. Tobin, (1958), 'Estimation of relationships for limited dependent variables', Econometrica 26, 24-36.
11. JS. Job,, N.A. Halsey,, R. Boulos,, E. Holt,, D. Farrell,, P. Albrecht,, JR. Brutus,, M. Adrien,, J. Andre,, E. Chan,, P. Kissinger,, C. Boulos, & Project Team. the CiteSoleil/JHU, (1991), 'Successful immunization of infants at 6 months of age with high dose Edmonston-Zagreb measles vaccine', Pediatric Infectious Diseases Journal 10, 303-311.
12. L. H. Ahrens, (1954), Quantitative Spectrochemical Analysis of Silicates, London, Pergamon Press.
13. L. Moulton, & N.A. Halsey, (1995), 'A Mixture Model with Detection Limits for Regression Analyses of Antibody Response to Vaccine', Biometrics 51, 1570-1578.
14. L. Moulton, & N.A. Halsey, (1996), 'A Mixed Gamma Model for Regression Analyses of Quantitative Assay Data', Vaccine 14, 1154-1158.
15. R. B. Arellano-Valle, & A. Azzalini, (2008), 'The centred parametrization for the multivariate skew-normal distribution', Journal of Multivariate Analysis 99, 1362-1382.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv37n1a12,
AUTHOR = {Martínez-Flórez, Guillermo and Barrera, Carlos},
TITLE = {{Asymmetric Regression Models Bernoulli/LogProportional-Hazard Distribution}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2014},
volume = {37},
number = {1},
pages = {183-198}
}
References
Ahrens, L. H. (1954), Quantitative Spectrochemical Analysis of Silicates, London, Pergamon Press.
Akaike, H. (1974), ‘A new look at statistical model identification’, IEEE Transaction on Automatic Control AU-19, 716–722.
Arellano-Valle, R. B. & Azzalini, A. (2008), ‘The centred parametrization for the multivariate skew-normal distribution’, Journal of Multivariate Analysis 99, 1362–1382.
Azzalini, A. (1985), ‘A class of distributions which includes the normal ones’, Scandinavian Journal of Statistics 12, 171–178.
Azzalini, A., dal Cappello, T. & Kotz, S. (2003), ‘Log-skew-normal and log-skew-t distributions as models for family income data’, Journal of Income Distribution 11, 12–20.
Chai, H. & Bailey, K. (2008), ‘Use of log-normal distribution in analysis of continuous data with a discrete component at zero’, Statistics in Medicine 27, 3643–3655.
Cragg, J. (1971), ‘Some statistical models for limited dependent variables with application to the demand for durable goods’, Econometrica 39, 829–844.
Job, J., Halsey, N., Boulos, R., Holt, E., Farrell, D., Albrecht, P., Brutus, J., Adrien, M., Andre, J., Chan, E., Kissinger, P., Boulos, C. & the CiteSoleil/JHU, P. T. (1991), ‘Successful immunization of infants at 6 months of age with high dose edmonston-zagreb measles vaccine’, Pediatric Infectious Diseases Journal 10, 303–311.
Martínez-Florez, G., Bolfarine, H. & Gómez, H.W. (2013), ‘Asymmetric regression models with limited responses with an application to antibody response to vaccine’, Biometrical Journal 55, 156–172.
Martínez-Florez, G., Moreno-Arenas, G. & Vergara-Cardozo, S. (2013), ‘Properties and inference for proportional hazard models’, Revista Colombiana de Estadística 36(1), 95–114.
Mateu-Figueras, G. & Pawlosky-Glanh (2003), Una alternativa a la distribución log-normal, in ‘Actas del XXVII Congreso Nacional de Estadística e Investigación Operativa (SEIO)’, Sociedade de Estadítica e Investigación Operativa, España, pp. 1849–1858.
Mateu-Figueras, G., Pawlosky-Glanh. & Barcelo-Vidal, C. (2004), The natural law in geochemistry: Lognormal or log skew-normal?, in ‘32th International Geological Congress’, International Union of Soil Sciences, Florence, Italy, pp. 1849–1858.
Moulton, L. & Halsey, N. (1995), ‘A mixture model with detection limits for regression analyses of antibody response to vaccine’, Biometrics 51, 1570–1578.
Moulton, L. & Halsey, N. (1996), ‘A mixed Gamma model for regression analyses of quantitative assay data’, Vaccine 14, 1154–1158.
Tobin, J. (1958), ‘Estimation of relationships for limited dependent variables’, Econometrica 26, 24–36.
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