*1.1. Asymmetry*

Earlier results on asymmetric models started with the pioneering works by [1,3,4]. This topic regained interest with the study in [5], which from a Bayesian point of view developed a new asymmetric model which was later studied in depth by Azzalini [6], from a classical point of view. Azzalini model was termed the skew-normal distribution. Following Azzalini's method, a general family of asymmetric models termed skew-symmetric models appeared in the literature. The following lemma, originally presented in [6], can be considered as the starting point for the development of these asymmetric models.

**Lemma 1.** *Let f*0 *be a probability density function (pdf) which is symmetric around zero, and G a cumulative distribution function (cdf) such that G exists and is a symmetric pdf around zero. Then*

$$f\_{\mathbb{Z}}(z;\lambda) = 2f\_0(z)G(\lambda z), \quad z \in \mathbb{R}, \tag{1}$$

*is a pdf for λ* ∈ R*.*

Equation (1) provides the skew version of *f*0(·) with skewing function *<sup>G</sup>*(·) and *λ* the skewness parameter. If *f*0(·) = *φ*(·) and *<sup>G</sup>*(·) = <sup>Φ</sup>(·), the pdf and cdf, respectively, of the *N*(0, 1) distribution, then the skew-normal is obtained, whose pdf is

$$f\_Z(z) = 2\phi(z)\Phi(\lambda z), \qquad z \in \mathbb{R}, \ \lambda \in \mathbb{R} \tag{2}$$

Other examples of skew models are: skew-t, skew-Cauchy, skew-elliptical, and generalized skew-elliptical. We highlight that all of them are unimodal distributions.
