Properties and Inference for Proportional Hazard Models

Propiedades e inferencia para modelos de Hazard proporcional

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 asymmetryor 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.

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.

Properties and Inference for Proportional Hazard Models Propiedades e inferencia para modelos de Hazard proporcional GUILLERMO MARTÍNEZ-FLÓREZ¹, GERMÁN MORENO-ARENAS², SANDRA VERGARA-CARDOZO³

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

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Abstract

We consider an arbitrary continuous cumulative distribution function F(x) with a probability density function f(x) = dF(x)/dx and hazard function h_f(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 h_f(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.

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Resumen

Palabras clave: asimetría, curtosis, distribución skew-normal, función de riesgo, método de los momentos, modelo de riesgo proporcional, verosimilitud perfilada.

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[Recibido en junio de 2012. Aceptado en abril de 2013]

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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}
}

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