The Extended Skew-Exponential Power Distribution and Its Derivation

Abstract

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.

Keywords:

• Kurtosis
• Maximum likelihood estimation
• Singular information matrix
• Skewness

Mathematics Subject Classification:

• Primary 60E05
• Secondary 62F30

Acknowledgments

We are most grateful to the anonymous referees for their comments which enhanced the presentation of the article. We would also like to express our thanks to the Guest Editors of the Special Issue, Graciela González Farías and Arthur Pewsey, for insightful comments on, and informative and stimulating discussions concerning, the manuscript. Finally, we gratefully acknowledge partial financial support in the form of grants MECESUP PUC-0103 (Salinas), FONDECYT 1040865/1060727 (Arellano-Valle), and FONDECYT 1060727 (Gómez).