¹Universidad de Sonora, División de Ciencias Exactas y Naturales, Departamento de Matemáticas, Hermosillo, México. Professor. Email: montoya@mat.uson.mx
²Universidad de Sonora, División de Ciencias Exactas y Naturales, Departamento de Matemáticas, Hermosillo, México. Professor. Email: gfiguero@gauss.mat.uson.mx
³Institute of Metals and Technology, Ljubljana, Slovenia. Research assistant. Email: nusa.puksic@imt.si

This paper concerns to the problem of making inferences about the vulnerability θ=P(X\textgreater v) and the mixing proportion p parameters, when the random variable X is distributed as a mixture of two Gumbel distributions and v is a known fixed value. A profile likelihood approach is proposed for the estimation of these parameters. This approach is a powerful though simple method for separately estimating a parameter of interest in the presence of unknown nuisance parameters. Inferences about θ, p or \left(θ,p\right) are given in terms of profile likelihood regions and can be easily obtained on a computer. This methodology is illustrated through a real problem where the main purpose is to model the size of non-metallic inclusions in steel.

Key words: Invariance principle, Likelihood approach, Likelihood region, Mixture of distributions.

En este artículo consideramos el problema de hacer inferencias sobre el parámetro de vulnerabilidad θ=P(X\textgreater v) y la proporción de mezcla p cuando X es una variable aleatoria cuya distribución es una mezcla de dos distribuciones Gumbel y v es un valor fijo y conocido. Se propone el enfoque de verosimilitud perfil para estimar estos parámetros, el cual es un método simple, pero poderoso, para estimar por separado un parámetro de interés en presencia de parámetros de estorbo desconocidos. Las inferencias sobre θ, p o \left(θ,p\right) se presentan por medio de regiones de verosimilitud perfil y se pueden obtener fácilmente en una computadora. Esta metodología se ilustra mediante un problema real donde se modela el tamaño de inclusiones no metálicas en el acero.

Palabras clave: enfoque de verosimilitud, mezcla de distribuciones, principio de invarianza, región de verosimilitud.

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