*2.2. Statistical Properties*

### 2.2.1. Shape of the Density

The MPN distribution exhibits a bell-shaped form, which can be symmetric or positively or negatively skewed depending on the value of the parameter *α*. Now, we derive some analytical expressions that are useful to obtain approximations of modal values and inflection points of this model. In the following, it will be assumed that *μ* = 0 and *σ* = 1.

**Proposition 1.** *The pdf of X* ∼ MPN (0, 1, *α*) *has a local maximum at* (*<sup>x</sup>*1, *f*(*x*1; *α*)) *and two inflection points at* (*<sup>x</sup>*2, *f*(*x*2; *α*)) *and* (*<sup>x</sup>*3, *f*(*x*3; *<sup>α</sup>*))*, respectively, where x*1 *is the root of the equation*

$$\mathbf{x}^\* = \frac{(\alpha - 1)\phi(\mathbf{x}^\*)}{1 + \Phi(\mathbf{x}^\*)},\tag{9}$$

*and x*2 *and x*3 *are two solutions of the equation*

$$1 = \left( -\mathbf{x} + \frac{(\boldsymbol{\alpha} - \mathbf{1})\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \right)^2 - \frac{(\boldsymbol{\alpha} - \mathbf{1})\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \left( \mathbf{x} + \frac{\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \right). \tag{10}$$

**Proof.** The proof consists of simple derivatives of the function *f* . From the equation (8), we calculate

$$\frac{\partial}{\partial \mathbf{x}} f(\mathbf{x}; a) \quad = \quad \frac{a}{2^{a} - 1} \phi(\mathbf{x}) [1 + \Phi(\mathbf{x})]^{a - 1} \left( -\mathbf{x} + \frac{(a - 1)\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \right).$$

$$\frac{\partial^{2}}{\partial \mathbf{x}^{2}} f(\mathbf{x}; a) \quad = \quad \frac{a}{2^{a} - 1} \phi(\mathbf{x}) [1 + \Phi(\mathbf{x})]^{a - 1} \left\{ \left( -\mathbf{x} + \frac{(a - 1)\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \right)^{2} - \left[ 1 + \frac{(a - 1)\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \right]^{2} \right\}$$

$$\times \left( \mathbf{x} + \frac{\phi(\mathbf{x})}{1 + \Phi(\mathbf{x})} \right) \Big| \, \cdot \,.$$

By setting Equations (9) and (10) to be equal to zero, the results are obtained after some algebra. Figure 2 displays the graph of the first derivative of *f*(·), where it is observed that the maximum exists and it is unique. Therefore, the MPN distribution is unimodal. -

**Figure 2.** Plot of the first derivative of MPN distribution for selected values of the parameters.

**Table 1.** Approximations

 of the roots of

**Remark 3.** *The solutions of Equations (9) and (10) can be numerically obtained by using the built-in function "uniroot" in the software package* R*. Table 1 below illustrates some approximations of the roots x*1*, x*2*, and x*3*, and the corresponding figures of the pdf evaluated at these values.*

Equationscorresponding figures of the pdf of the MPN evaluated at these roots. *α x***1** *x***2** *x***3** *f* **(***x***1;** *α***)** *f* **(***x***2;** *α***)** *f* **(***x***3;** *α***)**

 (9) and (10) for some values of *α*, and the

