*2.4. Moments*

In this sub-section, the following proposition shows the computation of the moments of a random variable *T* ∼ *MSHN*(*<sup>σ</sup>*, *q*). Hence, it also displays the coefficients of asymmetry and kurtosis.

**Proposition 5.** *Let T* ∼ MSHN(*<sup>σ</sup>*, *q*)*. Then the r-th moment of T is given by*

$$\mu\_r = \mathbb{E}(T^r) = \frac{2^{r\left(\frac{1}{q} + \frac{1}{2}\right)}}{\sqrt{\pi}} \sigma^r \Gamma\left(\frac{r+1}{2}\right) \Gamma\left(\frac{q-r}{q}\right), \ q > r,\tag{8}$$

*where* <sup>Γ</sup>(·) *denotes the gamma function.*

**Proof.** Let *W* ∼ *Wei*(*q*, 1/2) and using Proposition 3, we have

$$\mu\_r = E(T') = E\left(E(X'|\mathcal{W}')\right) = E\left(\sqrt{\frac{2^r}{\pi}}\Gamma\left(\frac{r+1}{2}\right)\sigma^r \mathcal{W}^{-r}\right) = \sqrt{\frac{2^r}{\pi}}\Gamma\left(\frac{r+1}{2}\right)\sigma^r E\left(\mathcal{W}^{-r}\right),$$

where *<sup>E</sup>*(*<sup>W</sup>*−*<sup>r</sup>*) = 2*r*/*q*Γ ((*q* − *<sup>r</sup>*)/*q*), *q* > *r* is the *r*-th moment of the inverse Weibull distribution .

**Corollary 1.** *Let T* ∼ MSHN(*<sup>σ</sup>*, *q*)*. Then the expectation and variance of T are given respectively by*

$$E(T) = \frac{2^{\frac{1}{q} + \frac{1}{2}}}{\sqrt{\pi}} \sigma \Gamma\left(\frac{q - 1}{q}\right), \ q > 1, \text{ and}$$

$$Var(T) = 2^{\left(\frac{2}{q} + 1\right)} \sigma^2 \left[\frac{1}{2} \Gamma\left(\frac{q - 2}{q}\right) - \frac{1}{\pi} \Gamma^2\left(\frac{q - 1}{q}\right)\right], \ q > 2.$$

**Corollary 2.** *Let T* ∼ MSHN(*<sup>σ</sup>*, *q*)*. Then the coefficients of asymmetry (β*1*) and kurtosis (β*2*) are given by*

$$\beta\_1 = \frac{\frac{1}{\sqrt{\pi}}\Gamma\left(\frac{q-3}{q}\right) - \frac{3}{2\sqrt{\pi}}\Gamma\left(\frac{q-1}{q}\right)\Gamma\left(\frac{q-2}{q}\right) + \frac{2}{\sqrt{\pi}}\Gamma^3\left(\frac{q-1}{q}\right)}{\left[\frac{1}{2}\Gamma\left(\frac{q-2}{q}\right) - \frac{1}{\pi}\Gamma^2\left(\frac{q-1}{q}\right)\right]^{3/2}}, \quad q > 3, \text{ and}$$

$$\beta\_2 = \frac{\frac{3}{4}\Gamma\left(\frac{q-4}{q}\right) - \frac{4}{\pi}\Gamma\left(\frac{q-1}{q}\right)\Gamma\left(\frac{q-3}{q}\right) + \frac{3}{\pi}\Gamma^2\left(\frac{q-1}{q}\right)\Gamma\left(\frac{q-2}{q}\right) - \frac{3}{\pi^2}\Gamma^4\left(\frac{q-1}{q}\right)}{\left[\frac{1}{2}\Gamma\left(\frac{q-2}{q}\right) - \frac{1}{\pi}\Gamma^2\left(\frac{q-1}{q}\right)\right]^2}, \quad q > 4.$$

**Remark 2.** *Figure 2 shows graphs of the coefficients of the MSHN distribution compared with those of the SHN distribution. Note that the MSHN distribution presents higher asymmetry and kurtosis values than the SHN distribution. Furthermore, in both distributions when q* → ∞ *the coefficients of asymmetry and kurtosis converge to* √2(4 − *π*)(*π* − <sup>2</sup>)−3/2 *and* (<sup>3</sup>*π*<sup>2</sup> − 4*π* − <sup>12</sup>)(*π* − <sup>2</sup>)−2*, respectively; they coincide with the coefficients of the HN distribution.*

**Figure 2.** Graph of the coefficients of asymmetry and kurtosis for the MSHN and SHN distributions.
