*2.2. Properties*

Next, important properties of the FBS model are presented. First an explicit expression for the cdf is given in terms of the cdf of a bivariante normal distribution.

**Proposition 1.** *Let T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*)*. Then the cdf of T is*

$$F\_{\Gamma}(t) = \begin{cases} \ c\_{\delta} \boldsymbol{\Phi}\_{\text{BN}\_{\lambda}} \left( \frac{\lambda \delta}{\sqrt{1+\lambda^{2}}}, a\_{l} - \delta \right), & \text{if } 0 < t < \beta \\\\ c\_{\delta} \left[ \boldsymbol{\Phi}\_{\text{BN}\_{\lambda}} \left( \frac{\lambda \delta}{\sqrt{1+\lambda^{2}}}, -\delta \right) + \boldsymbol{\Phi}\_{\text{BN}\_{\lambda}} \left( -\frac{\lambda \delta}{\sqrt{1+\lambda^{2}}}, a\_{l} + \delta \right) - \boldsymbol{\Phi}\_{\text{BN}\_{\lambda}} \left( -\frac{\lambda \delta}{\sqrt{1+\lambda^{2}}}, \delta \right) \right], & \text{if } t \ge \beta, \end{cases} \tag{9}$$

*where* <sup>Φ</sup>*BNλ* (*<sup>x</sup>*, *y*) *is the cdf of a bivariate normal distribution, with mean vector μ* = (0, 0) *and covariance matrix* 

$$
\Omega\_{\lambda} = \begin{pmatrix} 1 & \rho\_{\lambda} \\ \rho\_{\lambda} & 1 \end{pmatrix} \quad \text{where} \ \rho\_{\lambda} = -\frac{\lambda}{\sqrt{1+\lambda^2}} \ . \tag{10}
$$

**Proof.** It can be seen in Appendix A.

Next some particular cases of interest for *λ* and *δ* parameters are discussed. Results about the shape of *fT*(·) are included.

### 2.2.1. Effect of *λ*.

**Corollary 1.** *Let T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*)*. If λ* = 0 *then the cdf of T is*

$$F\_T(t) = \begin{cases} \frac{c\_\delta}{2} \Phi(a\_t - \delta), & \text{if } \begin{array}{l} 0 < t < \beta \\ \frac{c\_\delta}{2} \left\{ \Phi(a\_t + \delta) + 1 - 2\Phi(\delta) \right\}, & \text{if } \ t \ge \beta \end{array} \tag{11}$$

**Proof.** If *λ* = 0 then *ρλ*, defined in (10), is equal to zero, and since in the bivariate normal distribution uncorrelation implies independence, we have that

$$\Phi\_{BN\_{\lambda=0}}(\mathbf{x}, \mathbf{y}) \; := \Phi(\mathbf{x}) \; \Phi(\mathbf{y}) \; \; \qquad \forall (\mathbf{x}, \mathbf{y}) \; \; \; \theta$$

Taking into account that Φ(0) = 1/2 and <sup>Φ</sup>(−*<sup>δ</sup>*) = 1 − <sup>Φ</sup>(*δ*), we have that (9) reduces to (11).

Result in Corollary 1 corresponds to the model studied in [22].

### 2.2.2. Effect of *δ*.

**Corollary 2.** *Let T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*)*. If δ* = 0 *then FT reduces to FT*(*t*) = <sup>2</sup>Φ*BNλ*(0, *at*), *for t* > 0 . Corollary 2 is a particular case of models studied in [15].

2.2.3. Shape of *fT*(·).

**Proposition 2.** *Let T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*). *Then the pdf given in (*8*) is nondifferentiable at t* = *β*.

**Proof.** It follows from (8), by noting that if *t* = *β* then *at* = 0 and the absolute value function is not differentiable at zero.

**Proposition 3.** *Let T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*). *The pdf given in (*8*) can be bimodal. The modes are the solution of the following non-linear equations.*

*1.* 0 < *t*∗1< *β solution of*

$$a\_{t\_1} = \delta \, + \, \lambda \frac{\phi(\lambda a\_{t\_1})}{\Phi(\lambda a\_{t\_1})} + \frac{a\_{t\_1}^{\prime\prime}}{\left\{a\_{t\_1}^{\prime}\right\}^2} \,. \tag{12}$$

*2. t*∗2> *β solution of*

$$a\_{t\_2} = -\delta + \lambda \frac{\Phi(\lambda a\_{t\_2})}{\Phi(\lambda a\_{t\_2})} + \frac{a\_{t\_2}^{\prime\prime}}{\left\{a\_{t\_2}^{\prime}\right\}^2} \; , \tag{13}$$

*With at given in (6), a tand a tthe first and second derivatives of at with respect to t, respectively.*

### **Proof.** It is given in Appendix A.

Comments on the use of (12) and (13) are included in Appendix A, Remark A1.

**Remark 1.** *Equations obtained in (12) and (13) are similar to those we have in the skew normal and BS model.*

*1. Let Z* ∼ *SN*(*λ*)*, λ* ∈ R*. Then Z is unimodal and the mode, <sup>z</sup>*<sup>∗</sup>*, is given by the solution of the non-linear equation*

$$z = \lambda \frac{\Phi(\lambda z)}{\Phi(\lambda z)} \dots$$

*2. Let T* ∼ *BS*(*<sup>α</sup>*, *β*)*, α*, *β* > 0. *Then T is unimodal and the mode, t*<sup>∗</sup>*, is given by the solution of the non-linear equation*

$$-a\_t \left\{ a\_t' \right\}^2 + a\_t'' = 0.$$

Next it is shown that the *p*-th quantile of *T* can be given in terms of the *pth* quantile of the *FSN*(*<sup>δ</sup>*, *<sup>λ</sup>*). Also it is proved that the FBS model is closed under change of scale and reciprocity.

**Theorem 1.** *Let T* ∼ *FBS*(*<sup>α</sup>*, *β*, *δ*, *<sup>λ</sup>*)*, with α*, *β* ∈ R<sup>+</sup> *and δ*, *λ* ∈ R*. Then*

*(i) Let tp be the pth quantile of T ,* 0 < *p* < 1*.*

$$t\_p = \beta \left(\frac{\alpha}{2} z\_p + \sqrt{\left(\frac{\alpha}{2} z\_p\right)^2 + 1}\right)^2 \tag{14}$$

*where zp denotes the pth quantile of Z* ∼ *FSN*(*<sup>δ</sup>*, *<sup>λ</sup>*).


**Proof.** It can be seen in Appendix A.
