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Properties and Inference for Proportional Hazard Models
Propiedades e inferencia para modelos de Hazard proporcional
Keywords:
Hazard function, Kurtosis, Method of moments, Profile likelihood, Proportional hazard model, Skewness, Skew-normal distribution (en)Asimetría, curtosis, distribución skew-normal, función de riesgo, método de los momentos, modelo de riesgo proporcional, verosimilitud perfilada. (es)
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Consideramos una función de distribución continua arbitraria F(x) con función de densidad de probabilidad f(x) = dF(x)=dx y función de riesgo hf (x) = f(x)=[1 F(x)]: En este artículo proponemos una nueva familia de distribuciones cuya función de riesgo es proporcional a la función de riesgo hf (x). El modelo propuesto puede ajustar datos con alta asimetría o curtosis fuera del rango de cobertura permitido por la distribución normal, t-Student, logística, entre otras. Estimamos los parámetros del modelo usando máxima verosimilitud, verosimilitud perfilada y el método elemental de percentiles. Calculamos las matrices de información esperada y observada. Consideramos test de verosimilitudes para algunas hipótesis de interés en el modelo con función de riesgo proporcional a la distribución normal. Presentamos una aplicación con datos reales que ilustra que el modelo propuesto es adecuado.
1Universidad de Córdoba, Facultad de Ciencias, Departamento de Matemáticas y Estadística, Montería, Colombia. Professor. Email: gmartinez@correo.unicordoba.edu.co
2Universidad Industrial de Santander, Facultad de Ciencias, Escuela de Matemáticas, Bucaramanga, Colombia. Associate professor. Email: gmorenoa@uis.edu.co
3Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Bogotá, Colombia. Assistant Professor. Email: svergarac@unal.edu.co
We consider an arbitrary continuous cumulative distribution function F(x) with a probability density function f(x) = dF(x)/dx and hazard function hf(x)=f(x)/[1-F(x)]. We propose a new family of distributions, the so-called proportional hazard distribution-function, whose hazard function is proportional to hf(x). The new model can fit data with high asymmetry or kurtosis outside the range covered by the normal, t-student and logistic distributions, among others. We estimate the parameters by maximum likelihood, profile likelihood and the elemental percentile method. The observed and expected information matrices are determined and likelihood tests for some hypotheses of interest are also considered in the proportional hazard normal distribution. We show an application to real data, which illustrates the adequacy of the proposed model.
Key words: Hazard function, Kurtosis, Method of moments, Profile likelihood, Proportional hazard model, Skewness, Skew-normal distribution.
Consideramos una función de distribución continua arbitraria F(x) con función de densidad de probabilidad f(x)=dF(x)/dx y función de riesgo hf(x)=f(x)/[1-F (x)]. En este artículo proponemos una nueva familia de distribuciones cuya función de riesgo es proporcional a la función de riesgo hf(x). El modelo propuesto puede ajustar datos con alta asimetría o curtosis fuera del rango de cobertura permitido por la distribución normal, t-Student, logística, entre otras. Estimamos los parámetros del modelo usando máxima verosimilitud, verosimilitud perfilada y el método elemental de percentiles. Calculamos las matrices de información esperada y observada. Consideramos test de verosimilitudes para algunas hipótesis de interés en el modelo con función de riesgo proporcional a la distribución normal. Presentamos una aplicación con datos reales que ilustra que el modelo propuesto es adecuado.
Palabras clave: asimetría, curtosis, distribución skew-normal, función de riesgo, método de los momentos, modelo de riesgo proporcional, verosimilitud perfilada.
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References
1. Akaike, H. (1974), 'A new look at statistical model identification', IEEE Transaction on Automatic Control AC-19(6), 716-722.
2. Arellano-Valle, R. B., Gómez, H. W. & Quintana, F. (2004), 'A new class of skew-normal distributions', Communications in Statistics - Theory and Methods 33, 1465-1480.
3. Arellano-Valle, R. B., Gómez, H. W. & Quintana, F. (2005), 'Statistical inference for a general class of asymmetric distributions', Journal of Statistical Planning and Inference 128, 427-443.
4. Arnold, B., Gómez, H. & Salinas, H. (2009), 'On multiple constraint skewed models', Statistics 43-3, 279-293.
5. Azzalini, A. (1985), 'A class of distributions which includes the normal ones', Scandinavian Journal of Statistics 12, 171-178.
6. Barndorff-Nielsen, O. (1983), 'On a formula for the distribution of the maximum likelihood estimator', Biometrika 70(2), 343-365.
7. Castillo, E. & Hadi, A. (1995), 'A method for estimating parameters and quantiles of distributions of continuous random variables', Computational Statistics and Data Analysis 20(4), 421-439.
8. Chiogna, M. (1998), 'Some results on the scalar skew-normal distribution', Journal of the Italian Statistical Society 1, 1-14.
9. Durrans, S. R. (1992), 'Distributions of fractional order statistics in hydrology', Water Resources Research 28-6, 1649-1655.
10. Efron, B. (1979), 'Bootstrap methods: another look at the Jackknife', Annals of Statistics 7, 1-26.
11. Efron, B. (1982), 'The Jackknife, the Bootstrap, and other Resampling Plans', CBMS 38, SIAM-NSF.
12. Efron, B. & Tibshirani, R. J. (1993), An Introduction to the Bootstrap, Chapman and Hall, New York.
13. Eugene, N., Lee, C. & Famoye, F. (2002), 'Beta-normal distribution and its applications', Communications in Statistics-Theory and Methods 31, 497-512.
14. Farias, R., Moreno, G. & Patriota, A. (2009), 'Reducción de modelos en la pre-sencia de parámetros de perturbación', Revista Colombiana de Estadística 32(1), 99-121.
15. Fernández, C. & Steel, M. F. J. (1998), 'On Bayesian modeling of fat tails and skewness', Journal of the American Statistical Association 93, 359-371.
16. Gupta, A. K., Chang, F. C. & Huang, W. J. (2002), 'Some skew-symmetric models', Random Operators Stochastic Equations 10, 113-140.
17. Gupta, D. & Gupta, R. (2008), 'Analyzing skewed data by power normal model', Test 17, 197-210.
18. Gómez, H., Venegas, O. & Bolfarine, H. (2007), 'Skew-symmetric distributions generated by the distribution function of the normal distribution', Environmetrics 18, 395-407.
19. Henze, N. (1986), 'A probabilistic representation of the skew-normal distribution', Scandinavian Journal of Statistics 13, 271-275.
20. Lehmann, E. L. (1953), 'A graphical estimation of mixed Weibull parameter in life testing electron tubes', Technometrics 1, 389-407.
21. Mudholkar, G. S. & Hutson, A. D. (2000), 'The epsilon-skew-normal distribution for analyzing near-normal data', Journal of Statistical Planning and Inference 83, 291-309.
22. O'Hagan, A. & Leonard, T. (1976), 'Bayes estimation subject to uncertainty about parameter constraints', Biometrika 63, 201-203.
23. Pewsey, A. (2000), 'Problems of inference for Azzalini's skewnormal distribution', Journal of Applied Statistics 27-7, 859-870.
24. Pewsey, A., Gómez, H. & Bolfarine, H. (2012), Likelihood based inference for distributions of fractional order statistics., 'II Jornada Internacional de Probabilidad y Estadística', Pontificia Universidad Católica del Perú, , .
25. R Development Core Team, (2013), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *http://www.R-project.org
26. Roberts, C. (1966), 'A correlation model useful in the study of twins', Journal of the American Statistical Association 61, 1184-1190.
27. Sen, P. & Singer, J. (1993), Large Sample Methods in Statistics: An Introdution with Applications, Chapman and Hall, New York.
28. Sen, P., Singer, J. & Pedroso de Lima, A. (2010), From Finite Sample to Asymptotic Methods in Statistics, Cambridge Series in Statistical and Probabilistic Mathematics.
29. Severini, T. A. (2000), Likelihood Methods in Statistics, Oxford University press.
30. Severini, T. (1998), 'An approximation to the modified profile likelihood function', Biometrika 85, 403-411.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv36n1a06,
AUTHOR = {Martínez-Flórez, Guillermo and Moreno-Arenas, Germán and Vergara-Cardozo, Sandra},
TITLE = {{Properties and Inference for Proportional Hazard Models}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2013},
volume = {36},
number = {1},
pages = {95-114}
}
References
Akaike, H. (1974), ‘A new look at statistical model identification’, IEEE Transaction on Automatic Control AU–19, 716–722.
Arellano-Valle, R. B., Gómez, H. W. & Quintana, F. (2004), ‘A new class of skewnormal distributions’, Communications in Statistics – Theory and Methods 33, 1465–1480.
Arellano-Valle, R. B., Gómez, H. W. & Quintana, F. (2005), ‘Statistical inference for a general class of asymmetric distributions’, Journal of Statistical Planning and Inference 128, 427–443.
Arnold, B., Gómez, H. & Salinas, H. (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, 171–178.
Barndorff-Nielsen, O. (1983), ‘On a formula for the distribution of the máximum likelihood estimator’, Biometrika 70(2), 343–365.
Castillo, E. & Hadi, A. (1995), ‘A method for estimating parameters and quantiles of distributions of continuous random variables’, Computational Statistics and Data Analysis 20(4), 421–439.
Chiogna, M. (1998), ‘Some results on the scalar skew-normal distribution’, Journal of the Italian Statistical Society 1, 1–14.
Durrans, S. R. (1992), ‘Distributions of fractional order statistics in hydrology’, Water Resources Research 28–6, 1649–1655.
Efron, B. (1979), ‘Bootstrap methods: another look at the Jackknife’, Annals of Statistics 7, 1–26.
Efron, B. (1982), ‘The Jackknife, the Bootstrap, and other Resampling Plans’, CBMS 38, SIAM-NSF .
Efron, B. & Tibshirani, R. J. (1993), An Introduction to the Bootstrap, Chapman and Hall, New York.
Eugene, N., Lee, C. & Famoye, F. (2002), ‘Beta-normal distribution and its applications’, Communications in Statistics – Theory and Methods 31, 497–512.
Farias, R., Moreno, G. & Patriota, A. (2009), ‘Reducción de modelos en la presencia de parámetros de perturbación’, Revista Colombiana de Estadística 32(1), 99–121.
Fernandez, C. & Steel, M. (1998), ‘On Bayesian modeling of fat tails and skewness’, Journal of the American Statistical Association 93–441, 359–371.
Gómez, H., Venegas, O. & Bolfarine, H. (2007), ‘Skew-symmetric distributions generated by the distribution function of the normal distribution’, Environmetrics 18, 395–407.
Gupta, A. K., Chang, F. C. & Huang, W. J. (2002), ‘Some skew-symmetric models’, Random Operators Stochastic Equations 10, 113–140.
Gupta, D. & Gupta, R. (2008), ‘Analyzing skewed data by power normal model’, Test 17, 197–210.
Henze, N. (1986), ‘A probabilistic representation of the skew-normal distribution’, Scandinavian Journal of Statistics 13, 271–275.
Lehmann, E. L. (1953), ‘A graphical estimation of mixed Weibull parameter in life testing electron tubes’, Technometrics 1, 389–407.
Mudholkar, G. S. & Hutson, A. D. (2000), ‘The epsilon-skew-normal distribution for analyzing near-normal data’, Journal of Statistical Planning and Inference 83, 291–309.
O’Hagan, A. & Leonard, T. (1976), ‘Bayes estimation subject to uncertainty about parameter constraints’, Biometrika 63, 201–203.
Pewsey, A. (2000), ‘Problems of inference for Azzalini’s skewnormal distribution’, Journal of Applied Statistics 27–7, 859–870.
Pewsey, A., Gómez, H. & Bolfarine, H. (2012), Likelihood based inference for distributions of fractional order statistics., in ‘II Jornada Internacional de Probabilidad y Estadística’, Pontificia Universidad Católica del Perú.
R Development Core Team, R. (2013), ‘R: A language and environment for statistical computing’, R Foundation for Statistical Computing, Vienna, Austria.ISBN 3-900051-07-0.
Roberts, C. (1966), ‘A correlation model useful in the study of twins’, Journal of the American Statistical Association 61, 1184–1190.
Sen, P. & Singer, J. (1993), Large Sample Methods in Statistics: An Introdution with Applications, Chapman and Hall, New York.
Sen, P., Singer, J. & Pedroso de Lima, A. (2010), From Finite Sample to Asymptotic Methods in Statistics, Cambridge Series in Statistical and Probabilistic Mathematics.
Severini, T. (1998), ‘An approximation to the modified profile likelihood function’, Biometrika 85, 403–411.
Severini, T. A. (2000), Likelihood Methods in Statistics, Oxford University press.
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