Flexible Birnbaum–Saunders Distribution

Abstract

In this paper, we propose a bimodal extension of the Birnbaum-Saunders model by including an extra parameter. This new model is termed flexible Birnbaum-Saunders (FBS) and includes the ordinary Birnbaum-Saunders (BS) and the skew Birnbaum-Saunders (SBS) model as special cases. Its properties are studied. Parameter estimation is considered via an iterative maximum likelihood approach. Two real applications, of interest in environmental sciences, are included, which reveal that our proposal can perform better than other competing models.

Flexible Log-Linear Birnbaum–Saunders Model

“…In recent years, some improvements have been introduced to deal with this problem. In this sense, we highlight the flexible Birnbaum-Saunders model, proposed by Matínez-Flórez et al [2], where two parameters, δ ∈ R, λ > 0, are added to the usual Birnbaum-Saunders model, in such a way that λ controls asymmetry (skewness) and δ is a shape parameter related to unimodality/bimodality of our proposal.…”

“…Third, the flexible Log-linear Birnbaum-Saunders regression model is studied in Section 5. This can be used for non-negative lifetime variables following the flexible Birnbaum-Saunders model given in [2]. The model is built, particular cases of interest are cited, scores and maximum likelihood equations are given.…”

“…By applying the expression for the cdf of Z ∼ FSN(δ, λ) given in [2] we have that, for a γ,ξ,η (y) < 0, or, equivalently, for y < ξ…”

The Exponential-Centred Skew-Normal Distribution

“…Other distributions, though, have been proposed to model RT data. For example, it has been shown that the shifted Wald, shifted Birnbaum-Saunders [2] and shifted Gamma distributions [3] provide good fits to RT data (the Birnbaum-Saunders distribution has also been used to model neural activity [4], and a flexible version of it was proposed recently [5]. The slashed quasi-Gamma distribution (an extension of the generalised Gamma distribution) has been proposed to model data with varying skewness and high kurtosis [6].…”

Modified Power-Symmetric Distribution

“…For further details about the SN distribution, the reader is referred to Azzalini [4]. Martínez-Flórez et al [5] used generalizations of the SN distribution to extend the Birnbaum-Saunders model, and Contreras-Reyes and Arellano-Valle [6] utilized the Kullback-Leibler divergence measure to compare the multivariate normal distribution with the skew-multivariate normal. One of the main limitations of working with the family given by Equation (1) is that the information matrix could be singular for some of its particular models (see Azzalini [1]).…”