Abstract
In this paper, the skew-elliptical sinh-alpha-power distribution is developed as a natural follow-up to the skew-elliptical log-linear Birnbaum–Saunders alpha-power distribution, previously studied in the literature. Special cases include the ordinary log-linear Birnbaum–Saunders and skewed log-linear Birnbaum–Saunders distributions. As shown, it is able to surpass the ordinary sinh-normal models when fitting data sets with high (above the expected with the sinh-normal) degrees of asymmetry. Maximum likelihood estimation is developed with the inverse of the observed information matrix used for standard error estimation. Large sample properties of the maximum likelihood estimators such as consistency and asymptotic normality are established. An application is reported for the data set previously analyzed in the literature, where performance of the new distribution is shown when compared with other proposed alternative models.
Highlights
When observed data does not follow a normal distribution, the use of the elliptical family of distributions is an important alternative
A random variable (RV) X is distributed according to the elliptical distribution with location parameter ξ ∈ R and scale parameter η > 0
The null hypothesis is rejected and we conclude that the SPESN model fits the data better than the SHN model
Summary
When observed data does not follow a normal distribution, the use of the elliptical family of distributions is an important alternative. The class of elliptic distributions is a good alternative for situations of departure from normality, it is not appropriate when observations follow an asymmetric distribution These circumstances prompted the search for new distributions better suited to fit data with high asymmetry and kurtosis. Where H is an absolutely continuous distribution function with pdf h and α > 0 is a parameter that controls asymmetry and kurtosis of the distribution Barros et al [23] extended this model for the case of error distributions with heavier tails emphasizing the use of the Student t distribution They conducted estimation and diagnostic studies for the model entertained.
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