*2.3. Properties*

In this sub-section we study some properties of the MSHN distribution.

**Proposition 2.** *Let T* ∼ *MSHN*(*<sup>σ</sup>*, *q*)*, then when σ* = *q* = 1 *the density is*

$$f\_T(t) = \frac{4}{t^2} \left( \frac{1}{\sqrt{2\pi}} - \frac{2}{t} \exp(2/t^2) \Phi \left( -\frac{2}{t} \right) \right), \quad t > 0,\tag{5}$$

*where* <sup>Φ</sup>(·) *is the cdf of the standard normal.*

**Proof.** Using Proposition 1 for *σ* = *q* = 1, we have,

$$f\_{\Gamma}(t) = \frac{2}{\sqrt{2\pi}t^2} N\left(1, \frac{2}{z}, \frac{1}{2}, \frac{1}{2}\right) = \frac{2}{\sqrt{2\pi}t^2} \int\_0^\infty \exp\left(-\frac{2}{z}\mathbf{x}^{1/2} - \frac{1}{2v^2}\mathbf{x}\right) d\mathbf{x}, \quad t > 0. \tag{6}$$

Changing the variable *x* = *u*2 we obtain the result.

**Proposition 3.** *If T*|*W* = *w* ∼ *HN σw and Y*1/*q* = *W* ∼ *Wei*(*q*, 1/2) *then T* ∼ *MSHN*(*<sup>σ</sup>*, *q*)*.*

**Proof.** Since the marginal pdf of *T* is given by

$$f\_T(t; \sigma, q) = \int\_0^\infty f\_{T|W}(t|w) f\_W(w) \, dw = \frac{4q}{\sigma \sqrt{2\pi}} \int\_0^\infty w^q e^{-\frac{w^2 t^2}{2\sigma^2} - 2w^q} \, dw\_\sigma$$

and using the Lemma 1 in the Appendix A, the result is obtained.

**Proposition 4.** *Let T* ∼ *MSHN*(*<sup>σ</sup>*, *q*)*. If q* → ∞ *then T converges in law to a random variable T* ∼ *HN*(*σ*)*.*

**Proof.** Let *T* ∼ *MSHN*(*<sup>σ</sup>*, *q*) and *T* = *XY*1/*q* , where *X* ∼ *HN*(*σ*) and *Y* ∼ *Exp*(2). We study the convergence in law of *T*, since *Y* ∼ *Exp*(2) then *W* = *Y*1/*q* ∼ *Wei*(*q*, 1/2), we have that *E* (*W* − 1)2 = 1 22/*q* Γ(1 + 2/*q*) − 2 21/*q* Γ(1 + 1/*q*) + 1. If *q* → ∞ then *E* (*W* − 1)2 → 0, i.e., we have *W* P−→ 1 (see Lehmann [9]).

Since *T* ∼ *MSHN*(*<sup>σ</sup>*, *q*), by applying Slutsky's Lemma (see Lehmann [9]) to *T* = *XW* , we have

$$T \stackrel{\mathcal{L}}{\longrightarrow} X \sim HN(\sigma), \qquad \eta \to \infty,\tag{7}$$

that is, for increasing values of *q*, *T* converges in law to a *HN*(*σ*) distribution.

**Remark 1.** *Proposition 2 shows us that the MSHN*(1, 1) *distribution has a closed-form expression. Proposition 3 shows that an MSHN*(*<sup>σ</sup>*, *q*) *distribution can also be obtained as a scale mixture of an HN and a Wei distribution. This property is very important since it makes it possible to generate random numbers and implement the EM algorithm. Proposition 4 implies that, if q* → ∞ *then the cdf of an MSHN*(*<sup>σ</sup>*, *q*) *model approaches to the cdf of a HN*(*σ*) *distribution.*
