**3. Inference**

In this section, parameters estimation for the MPN distribution is discussed by using the method of moments and ML estimation. Additionally, a simulation analysis is carried out to illustrate the behavior of the ML estimators.

### *3.1. Method of Moments*

The following proposition illustrates the derivation of the moment estimates of the MPN distribution.

**Proposition 4.** *Let x*1, ... , *xn be a random sample obtained from the random variable X* ∼ MPN (*μ*, *σ*, *<sup>α</sup>*)*, then the moment estimates θM* = (*μM*, *σM*, *<sup>α</sup>M*) *for θ* = (*μ*, *σ*, *α*) *are given by*

$$
\hat{\sigma}\_M = \frac{S\_\mathbb{\hat\m}}{\sqrt{\frac{\hat\mu\_M}{2^{\hat\a}M} \left(a\_2(\hat{\alpha}\_M) - \frac{\hat\mu\_M}{2^{\hat\a}M} a\_1^2(\hat{\alpha}\_M)\right)}}, \quad \hat{\mu}\_M = \overline{\mathfrak{x}} - \hat{\sigma}\_M \frac{\hat\mu\_M}{2^{\hat\a}M} a\_1(\hat{\alpha}\_M) \tag{14}
$$

*and*

$$\frac{a\_3(\hat{\boldsymbol{a}}\_M) - \frac{3\hat{\boldsymbol{a}}\_M}{2^{\hat{\boldsymbol{a}}\_M} - 1} a\_1(\hat{\boldsymbol{a}}\_M) a\_2(\hat{\boldsymbol{a}}\_M) + \frac{2\hat{\boldsymbol{a}}\_M^2}{(2^{\hat{\boldsymbol{a}}\_M} - 1)^2} a\_1^3(\hat{\boldsymbol{a}}\_M)}{\left(\frac{\hat{\boldsymbol{a}}\_M}{2^{\hat{\boldsymbol{a}}\_M} - 1}\right)^{3/2} [a\_2(\hat{\boldsymbol{a}}\_M) - \frac{\hat{\boldsymbol{a}}\_M}{2^{\hat{\boldsymbol{a}}\_M} - 1} a\_1^2(\hat{\boldsymbol{a}}\_M)]^{3/2}} - A\_\pm = 0,\tag{15}$$

*where x, Sx and Ax denote the sample mean, sample standard deviation and sample Fisher's skewness coefficient respectively.*

**Proof.** As *μ* and *σ* are location and scale parameters respectively, the skewness coefficient does not depend on these parameters. Thus, the result in (15) is directly obtained from matching the sample skewness coefficient with population counterpart given in Corollary 2. In addition, by considering that *X* = *σZ* + *μ*, where *Z* ∼ MPN (0, 1, *<sup>α</sup>*), and again by equating sample mean and sample variance to the mean and variance respectively, it follows that

$$\begin{array}{rcl}\overline{\mathfrak{x}} &=& \widehat{\sigma}\_{M}\mathbb{E}(X) + \widehat{\mu}\_{M} \\\\ &=& \widehat{\sigma}\_{M}\frac{\widehat{\alpha}\_{M}}{2^{\widehat{\alpha}\_{M}}-1}a\_{1}(\widehat{\alpha}\_{M}) + \widehat{\mu}\_{M}.\end{array}$$

and

$$\begin{aligned} \mathcal{S}^2\_{\underline{\mathcal{K}}} &= \quad \widehat{\sigma}^2\_{M} \mathbb{V}ar(X) \\ &= \quad \widehat{\sigma}^2\_{M} \frac{\widehat{\alpha}\_{M}}{2^{\widehat{\alpha}\_{M}}-1} \left( a\_2(\widehat{\alpha}\_{M}) - \frac{\widehat{\alpha}\_{M}}{2^{\widehat{\alpha}\_{M}}-1} a\_1^2(\widehat{\alpha}\_{M}) \right) \end{aligned}$$

where *αM* satisfies expression (15). Then, (14) is obtained by solving the latter equations for *μM* and *σM*, respectively. -

### *3.2. Maximum Likelihood Estimation*

For a random sample *x*1, ... , *xn* derived from the MPN (*μ*, *σ*, *α*) distribution, the log-likelihood function can be written as

$$\ell(\mu, \sigma, \mathfrak{a}) = n\mathfrak{c}(\sigma, \mathfrak{a}) - \frac{1}{2\sigma^2} \sum\_{i=1}^{n} (\mathbf{x}\_i - \mu)^2 + (\mathfrak{a} - 1) \sum\_{i=1}^{n} \log \left[ 1 + \Phi\left(\frac{\mathbf{x}\_i - \mu}{\sigma}\right) \right],\tag{16}$$

where *<sup>c</sup>*(*<sup>σ</sup>*, *α*) = log(*α*) − log(2*α* − 1) − log(*σ*) − 12 log(<sup>2</sup>*π*).

The score equations are given by

$$n\mu + \sigma(\mathfrak{a} - 1)\sum\_{i=1}^{n} \kappa(\mathfrak{x}\_i) = n\overline{\mathfrak{x}}\_i \tag{17}$$

$$
\sigma^2 + \sigma(\alpha - 1) \sum\_{i=1}^n (\mathbf{x}\_i - \mu)\mathbf{x}(\mathbf{x}\_i) = \sum\_{i=1}^n (\mathbf{x}\_i - \mu)^2,\tag{18}
$$

$$\frac{n}{n} + \sum\_{i=1}^{n} \log\left[1 + \Phi\left(\frac{\mathbf{x}\_i - \mu}{\sigma}\right)\right] = \frac{2^n \log(2) n}{2^n - 1},\tag{19}$$

where *κ*(*w*) = *κ*(*w*; *μ*, *σ*) = *φ*( *<sup>w</sup>*−*μ σ* ) <sup>1</sup>+<sup>Φ</sup>( *<sup>w</sup>*−*μ σ* ).

Solutions for these Equations (17)–(19) can be obtained by using numerical procedures such as Newton–Raphson algorithm. Alternatively, these estimates can be found by directly maximizing the log-likelihood surface given by (16) and using the subroutine "optim" in the software package [21].
