*2.1. Probability Density Function*

**Definition 1.** *Let Z be a continuous and symmetric random variable with cdf <sup>G</sup>*(*z*; *η*) *and pdf g*(*z*; *η*)*, where η denotes a vector of parameters. We say that, a random variable, X, follows a* MPS *distribution, denoted as X* ∼ MPS(*η*, *<sup>α</sup>*)*, if its cdf is given by*

$$F(\mathbf{x}; \boldsymbol{\eta}, \boldsymbol{a}) = \frac{\left[1 + G(\mathbf{x}; \boldsymbol{\eta})\right]^{\boldsymbol{a}} - 1}{2^{\boldsymbol{a}} - 1},\tag{5}$$

*and its pdf is given by*

$$f(\mathbf{x}; \boldsymbol{\eta}, \boldsymbol{\alpha}) = \frac{\boldsymbol{\alpha}}{2^{\alpha} - 1} \, \mathrm{g}(\mathbf{x}; \boldsymbol{\eta}) \left[1 + \mathrm{G}(\mathbf{x}; \boldsymbol{\eta})\right]^{\alpha - 1}.\tag{6}$$

where *x* ∈ R and *α* > 0.

**Remark 1.** *In the case α* = 1*, the transformation given by Equation (6) is the identity. That is, the* MPS *distribution for α* = 1 *always provides the input probability density function.*

Thereforeforth, we proceed to examine the MPN distribution, whose cdf is provided by

$$F(\mathbf{x}; \mu, \sigma, \mathfrak{a}) = \frac{\left[1 + \Phi\left(\frac{\mathbf{x} - \mu}{\sigma}\right)\right]^a - 1}{2^a - 1},\tag{7}$$

and whose pdf is given by

$$f(\mathbf{x}; \mu, \sigma, \mathbf{a}) = \frac{a}{(2^a - 1)\sigma} \phi\left(\frac{\mathbf{x} - \mu}{\sigma}\right) \left[1 + \Phi\left(\frac{\mathbf{x} - \mu}{\sigma}\right)\right]^{a-1},\tag{8}$$

where *x* ∈ R, *μ* ∈ R is the location parameter, *σ* > 0 is the scale parameter, and *α* > 0 is the shape parameter. Hereafter, this will be denoted as *X* ∼ MPN (*μ*, *σ*, *<sup>α</sup>*). Figure 1 depicts some different shapes of the pdf of this model, for selected values of the parameter *α* with *μ* = −1, 1 and *σ* = 1. The MPN class of distributions is applicable for the change point problem, due to its favorable properties (see Maciak et al. [18]); moreover, the MPN model can be utilized in calibration (see Pesta [ ˇ 19]).

**Remark 2.** *Here, μ* ∈ R *and σ* > 0 *are location and scale parameters of the* MPN *distribution, respectively. For the particular case α* = 1*, these are not only location and scale parameters but also the mean and standard deviation of the standard normal distribution.*

**Figure 1.** Plot of the pdf of MPN distribution for selected values of the parameters.
