*1.2. Bimodality*

Another fundamental result in our proposal will be the following lemma, which was given in Gómez et al. [2]. These authors extended (1), by introducing a parameter *δ* in *f*0, in such a way that for certain values of *δ* the resulting distribution is bimodal.

**Lemma 2.** *Let f be a symmetric pdf around zero, F the corresponding cdf and G an absolutely continuous cdf such that G exists and is symmetric around zero. Then*

$$\mathcal{R}g(z;\delta,\lambda) = c\_{\delta}f(|z|+\delta)G(\lambda z), \qquad z \in \mathbb{R}, \quad \lambda, \delta \in \mathbb{R} \tag{3}$$

*is a pdf and c*<sup>−</sup><sup>1</sup> *δ* = 1 − *<sup>F</sup>*(*δ*).

Taking *f*(·) = *φ*(·) and *<sup>G</sup>*(·) = <sup>Φ</sup>(·), in (3), the flexible skew-normal (FSN) model was obtained and studied in detail in [2]. There, it was proved that the FSN model can be bimodal for certain values of *δ*. Notice that the FSN model is obtained by adding an extra parameter, *δ*, to the skew-normal distribution proposed in [6]. That is a random variable (rv) *Z* follows a FSN distribution, *Z* ∼ *FSN*(*<sup>δ</sup>*, *<sup>λ</sup>*), if its pdf is given by

$$f(z; \delta, \lambda) = c\_{\delta} \phi(|z| + \delta) \Phi(\lambda z), \qquad z \in \mathbb{R}, \ \lambda, \delta \in \mathbb{R} \tag{4}$$

where *φ* and Φ are the pdf and cdf of the *N*(0, 1) distribution, respectively, and *c*<sup>−</sup><sup>1</sup> *δ*= 1 − <sup>Φ</sup>(*δ*).

Other recent proposals in the contemporary literature dealing with bimodality are the extended two-pieces skew-normal model (ETN), introduced in [7] and the uni-bi-modal asymmetric power normal model given in [8] whose properties are based on results given in [9,10]. Applications of interest in Economics are given in [11]. All these references show the interest in the latest years for modelling bimodality.
