¹Universidad de La Frontera, Departamento de Matem\'{a}tica y Estad\'{\i}stica, Temuco, Chile. Professor. Email: asanhueza@ufro.cl
²Universidad de Valpara\'{\i}so, Departamento de Estad\'{\i}stica, CIMFAV, Valpara\'{\i}so, Chile. Professor. Email: victor.leiva@uv.cl
³Universidad Nacional de Colombia, Departamento de Estad\'{i}stica, Bogot\'{a}, Colombia. Assistant professor. Email: llopezk@unal.edu.co

In this article, we introduce a mixture inverse Gaussian (MIG) model based on the Student-t distribution and apply it to bacterium-based protein production for food industry. This model is mainly useful to describe data that follow positively skewed distributions and accommodate atypical observations in a better way than its classical version. Specifically, we present a characterization of the MIG-t distribution. In addition, we carry out a hazard analysis of this distribution centered mainly on its hazard rate. Furthermore, we discuss the maximum likelihood method, which produces--in this case--robust parameter estimates. Moreover, to evaluate the potential influence of atypical observations, we produce a diagnostic analysis for the model. Finally, we apply the obtained results to novel bacterium-based protein production data and statistically compare two types of protein producers using the likelihood ratio test based on the MIG-t model as an alternative methodology to the procedures available until now. This fact is very important, since the evaluation of protein production using both constructions allows practitioners to choose the most productive one before the bacterial culture is scaled to an industrial level.

Key words: Distribution mixture, Length-biased, Likelihood methods, distributions, R computer language.

En este art\iculo, introducimos un modelo Gaussiano inverso (MIG) mezclado basado en la distribuci\on t-Student y lo aplicamos a la producci\on de prote\inas basada en bacterias para la industria de alimentos. Este modelo es especialmente \util para describir datos que siguen una distribuci\on con sesgo positivo ya que permite acomodar observaciones at\ipicas de mejor forma que su versión cl\asica. Espec{i}ficamente, presentamos una caracterizaci\on de la distribución MIG-t y realizamos un an\alisis de confiabilidad de esta distribuci\on centrado principalmente en la tasa de fallas. También, discutimos el m\etodo de verosimilitud m\axima, el cual proporciona en este caso estimaciones robustas de los par\ametros del modelo. Con el fin de evaluar la influencia potencial de observaciones at\ipicas, proponemos un an\alisis de diagn\ostico para la distribuci\on. Finalmente, aplicamos los resultados obtenidos al análisis de datos nuevos de producci\on de prote\ina basada en bacterias utilizada en la industria de alimentos y comparamos estadísticamente dos tipos de bacterias productoras usando la prueba de raz\on de verosimilitudes basada en el modelo MIG-t como una metodolog\ia alternativa a los procedimientos disponibles a la fecha. Este punto es muy importante, ya que la evaluaci\on de producci\on de prote\inas usando dos construcciones distintas permite a los investigadores escoger el tipo m\as productivo antes de proceder al cultivo industrial a gran escala.

Palabras clave: distribuciones de largo sesgado, lenguaje de computaci\'{o}n R, m\'{e}todos de verosimilitud, mezcla de distribuciones.

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