*2.2. Density Function*

The following Proposition shows the pdf of the MSHN distribution with scale parameter *σ* and kurtosis parameter *q*, generated using the representation given in (3).

**Proposition 1.** *Let T* ∼ *MSHN*(*<sup>σ</sup>*, *q*)*. Then, the pdf of T is given by*

$$f\_{\Gamma}(t;\sigma,q) = \frac{2q}{\sqrt{2\pi\sigma^2}t^{q+1}}N\left(\frac{q+1}{2}, \frac{2}{t^q}, \frac{q}{2}, \frac{1}{2\sigma^2}\right),\tag{4}$$

*where t* > 0*, σ* > 0*, q* > 0*, and <sup>N</sup>*(·, ·, ·, ·) *is defined in Lemma 1 in the Appendix A.*

**Proof.** Using the stochastic representation given in (3) and the Jacobian method, we obtain that the density function associated with T is given by

$$f\_T(t; \sigma, q) \quad = \frac{4q}{\sqrt{2\pi\sigma^2}} \int\_0^\infty w^q \exp\left\{-\left(\frac{t^2 w^2}{2\sigma^2} + 2w^q\right)\right\} dw.$$

Making the change of variable *u* = *t*2*w*<sup>2</sup> we have,

$$\left(f\_T(t;\sigma,q)\right)\_t = \frac{2q}{\sqrt{2\pi\sigma^2 t^{q+1}}} \int\_0^\infty u^{\frac{q-1}{2}} \exp\left\{-\left(\frac{u}{2\sigma^2} + \frac{2u^{q/2}}{t^q}\right)\right\} du.$$

Hence, applying the Lemma 1 as set forth in the Appendix A, we obtain the result.

Figure 1 depicts plots of the density of the MSHN distribution for different values of parameter *q*.

**Figure 1.** The density function for different values of parameter *q* and *σ* = 1 in the MSHN distribution.

We perform a brief comparison illustrating that the tails of the MSHN distribution are heavier than those of the SHN distribution.

Table 1 shows the tail probability for different values in the SHN and MSHN models. It is immediately apparent that the MSHN tails are heavier than those of the SHN distribution.


**Table 1.** Tails comparison for different slashed half-normal (SHN) and modified slashed half-normal (MSHN) distributions.
