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Revista Colombiana de Estadística
Print version ISSN 0120-1751
Rev.Colomb.Estad. vol.45 no.2 Bogotá July/Dec. 2022 Epub Feb 01, 2023
https://doi.org/10.15446/rce.v45n2.95530
Artículos originales de investigación
Likelihood-Based Inference for the Asymmetric Exponentiated Bimodal Normal Model
Inferencia basada en verosimilitud para el modelo asimétrico bimodal normal exponenciado
1 Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería, Colombia
2 Facultad de Estadítica, Universidad Santo Tomás, Bogotá, Colombia
Asymmetric probability distributions have been widely studied by various authors in recent decades. Special interest has been had families of flexible distributions with the capability to have into account degree of skewness and kurtosis greater than the cl 1 distributions widely known in statistical theory. While, most of the new distributions fit unimodal data, and a few fit bimodal data, in the bimodal proposals, singularity problems have been found in the information matrices. Therefore, in this paper, extensions of the alpha-power family of distributions are developed, which have non-singular information matrix. The new proposals are based on the bimodal-normal and bimodal elliptical skew-normal distributions. These new extensions allow modeling asymmetric bimodal data, which are commonly found in several areas of scientific interest. The properties of these new distributions of probability are also studied in detail, and the statistical inference process is carried out to estimate the parameters of the proposed models. The stochastic convergence for the maximum likelihood estimator (MLE) vector can be found due to the non-singularity of the expected information matrix in the corresponding support. We also introduced extensions of the asymmetric bimodal normal and bimodal elliptical skew-normal models for the situations in which the data present censorship. A small simulation study to evaluate the properties of the MLE is also presented and, finally, two applications to real data set are presented for illustrative purposes.
Key words: Alpha-Power distribution; Asymmetric models; Bimodal normal distribution; Censored data; Maximum likelihood estimation
Las distribuciones de probabilidad asimétricas han sido ampliamente estudiadas por diversos autores en las últimas décadas. Se ha tenido especial interés en familias de distribuciones flexibles con la capacidad de tener en cuenta grados de asimetría y curtosis mayores que las distribuciones el ampliamente conocidas en teoría estadística. Si bien la mayoría de las nuevas distribuciones se ajustan a datos unimodales y unas pocas a datos bimodales, en las propuestas bimodales se han encontrado problemas de singularidad en las matrices de información. Por lo tanto, en este artículo se desarrollan extensiones de la familia de distribuciones alfa-potencia, que tienen matriz de información no singular. Las nuevas propuestas se basan en las distribuciones bimodal-normal y bimodal elíptica sesgada-normal. Estas nuevas extensiones permiten modelar datos bimodales asimétricos, que se encuentran comúnmente en varias áreas de interés científico. También se estudian en detalle las propiedades de estas nuevas distribuciones de probabilidad, y se realiza el proceso de inferencia estadística para estimar los parámetros de los modelos propuestos. La convergencia estocástica para el vector estimador de máxima verosimilitud (EMV) se puede encontrar debido a la no singularidad de la matriz de información esperada en el soporte correspondiente. También introdujimos extensiones de los modelos asimétrico bimodal normal y bimodal elíptico sesgado-normal para las situaciones en las que los datos presentan censura. También se presenta un pequeño estudio de simulación para evaluar las propiedades del EMV y, finalmente, se presentan dos aplicaciones a conjuntos de datos reales con fines ilustrativos.
Palabras clave: Distribución normal bimodal; Distribución alfa-potencia; Datos censurados; Modelos asimétricos; Estimación por máxima verosimilitud
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Received: June 2021; Accepted: February 2022
This is an open-access article distributed under the terms of the Creative Commons Attribution License
