*1.3. BS Model*

The BS or fatigue life distributions was proposed for modelling survival time data and material lifetime subject to stress in [12,13]. This model is asymmetric and only fits positive data. The pdf of a BS distribution is given by

$$f\_T(t) = \phi(a\_t) \frac{t^{-3/2}(t+\beta)}{2a\sqrt{\beta}}, \qquad t > 0,\tag{5}$$

where

$$a\_l = a\_l(a, \beta) = \frac{1}{a} \left( \sqrt{\frac{t}{\beta}} - \sqrt{\frac{\beta}{t}} \right),\tag{6}$$

*α* > 0 is a shape parameter, *β* > 0 is a scale parameter and the median of this distribution. (5) is denoted as *T* ∼ *BS*(*<sup>α</sup>*, *β*). It is well known that *α* is the parameter that controls asymmetry. Specifically, (5) becomes more asymmetric as *α* increases and symmetric around *β* as *α* gets close to zero. It can be seen in [13] that (5) can be obtained as the distribution of the random variable

$$T = \beta \left[ \frac{a}{2} Z + \sqrt{\left(\frac{a}{2} Z\right)^2 + 1} \right]^2,\tag{7}$$

where *Z* ∼ *N*(0, <sup>1</sup>).

The BS model has been applied to a variety of practical situations. However, quite often, although the data sugges<sup>t</sup> a BS distribution, some deficiencies are observed in the fitted BS model. This problem has motivated an increasing interest in its generalizations. We highlight that, recently, this model was extended by [14] to the family of elliptical distributions, this is known in the literature as the generalized Birnbaum–Saunders (GBS) distribution. Later, [15] proposed an extension based on the elliptical asymmetric distributions, known as the doubly generalized Birnbaum–Saunders model. On the other hand, [16] presents the asymmetric BS distribution with five parameters called the extended Birnbaum–Saunders (EBS) distribution. Other types of extensions are the asymmetric epsilon-Birnbaum–Saunders model given in [17], models in [18] based on the slash-elliptical family of distributions, and the generalized modified slash Birnbaum–Saunders (GMSBS) proposed in [19], which is based on [20].

In these extensions, we find that the asymmetric BS models previously cited, such as [15,21], are designed to fit data with greater or smaller asymmetry (or kurtosis) than that of the ordinary BS model, but they are not appropriate for fitting bimodal data. On the other hand, we highlight that the extension given in [21], which can become bimodal for certain combination of parameters is unable to capture bimodality unless it is accentuated enough.

Therefore there exists a real need for an asymmetric model, based on the BS distribution, and able to fit data presenting bimodal features, which is not uncommon in the literature. So the present paper presents a flexible BS distribution able to model skewness and to fit data with and without bimodality.

The paper is organized as follows. Section 2 is devoted to the development of an asymmetric uni-bimodal BS model. Its properties are studied in depth. Specifically, a closed expression for the cumulative distribution function (cdf) is given in terms of the cdf of a bivariate normal distribution. Some of the models proposed in [15,22] are obtained as particular cases. The shape and bimodality of the distribution are studied. It is shown that this model is closed under a change of scale and reciprocity. Survival and hazard functions are also obtained. Section 3 deals with moments derivation and iterative maximum likelihood estimation methods for the new model. Section 4 is devoted to real data applications of interest in environmental sciences. The first one deals with a bimodal situation in which our proposal performs better than other BS models and a mixture of normal distributions. The second one is taken from [16], where the extended BS model was proposed as the best for this dataset. It is shown that the FBS outperforms the extended BS model.

### **2. Results in Flexible Birnbaum-Saunders**

Based on the flexible skew-normal model proposed in [2], we extend the Birnbaum–Saunders. The main idea is to apply (7) with *Z* ∼ *FSN*(*<sup>δ</sup>*, *λ*) introduced in (4). This new model is called the flexible Birnbaum–Saunders (FBS) distribution whose pdf is given by

$$f(t; a, \beta, \delta, \lambda) = \frac{t^{-3/2}(t + \beta)}{2a\beta^{1/2}(1 - \Phi(\delta))}\phi(|a\_t| + \delta)\Phi(\lambda a\_t),\tag{8}$$

with *at* defined in (6), *t* > 0, *α* > 0, *β* > 0, *δ* ∈ R, *λ* ∈ R, *φ*(·) and <sup>Φ</sup>(·) the pdf and cdf of a *N*(0, <sup>1</sup>), respectively. We use the notation *T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*). The inclusion of parameters *δ* and *λ* makes our approach more flexible than the extensions previously discussed. *λ* is a parameter that controls asymmetry (skewness) and *δ* is a shape parameter related to bimodality of our proposal.

If *λ* = 0 then we obtain, as a particular case, the model introduced by [22].

Figures 1 and 2 depict the behaviour of (8) for some values of parameters, illustrating that it can be bimodal for some combinations of them.
