*3.3. Simulation Study*

To examine the behavior of the proposed approach, a simulation study is carried out to assess the performance of the estimation procedure for the parameters *μ*, *σ*, and *α* in the MPN model. The simulation analysis is conducted by considering 1000 generated samples of sizes *n* = 50, 100, and 200 from the MPN distribution. The goal of this simulation is to study the behavior of the ML estimators of the parameters by using our proposed procedure. To generate *X* ∼ MPN (*μ*, *σ*, *<sup>α</sup>*), the following algorithm is used,


where *μ* ∈ R, *σ* > 0, *α* > 0 and <sup>Φ</sup>−<sup>1</sup>(·) is the quantile function of the standard normal distribution. For each generated sample of the MPN distribution, the ML estimates and corresponding standard deviation (SD) were computed for each parameter. As it can be seen in Table 3, the performance of the estimates improves when *n* and *α* increases.

**Table 3.** Maximum likelihood (ML) estimates and standard deviation (SD) for the parameters *μ*, *σ* and *α* of the MPN model for different generated samples of sizes *n* = 50, 100, and 200.


Fisher's Information Matrix

Let us now consider *X* ∼ MPN (*μ*, *σ*, *α*) and *Z* = *X*−*μ σ* ∼ MPN (0, 1, *<sup>α</sup>*). For a single observation *x* of *X*, the log-likelihood function for *θ* = (*μ*, *σ*, *α*) is given by

$$\ell(\boldsymbol{\theta}) = \log f\_{\boldsymbol{X}}(\boldsymbol{\theta}, \boldsymbol{x}) = \boldsymbol{c}(\boldsymbol{\sigma}, \boldsymbol{a}) - \frac{1}{2\sigma^{2}}(\boldsymbol{x} - \boldsymbol{\mu})^{2} + (a - 1)\log\left[1 + \Phi\left(\frac{\boldsymbol{x} - \boldsymbol{\mu}}{\sigma}\right)\right].$$

The corresponding first and second partial derivatives of the log-likelihood function are derived in the Appendix A. It can be shown that the Fisher's information matrix for the MPN distribution is provided by

$$I\_F(\boldsymbol{\theta}) = \begin{pmatrix} I\_{\mu\mu} & I\_{\mu\sigma} & I\_{\mu\alpha} \\ & I\_{\sigma\sigma} & I\_{\sigma\alpha} \\ & & I\_{\alpha\alpha} \end{pmatrix}.$$

with the following entries,

$$\begin{array}{rcl} I\_{\mu\mu} & = & \frac{1}{\sigma^2} + \left(\frac{\mathfrak{a} - 1}{\sigma^2}\right)(b\_{11} + b\_{02}), \\\\ I\_{\mu\sigma} & = & \frac{2}{\sigma^2} \mathbb{E}(Z) - \left(\frac{\mathfrak{a} - 1}{\sigma^2}\right)(b\_{01} - b\_{21} - b\_{12}), \\\\ I\_{\mu\mu} & = & \frac{1}{\sigma} b\_{01}, \\\\ I\_{\sigma\tau} & = & -\frac{1}{\sigma^2} + \frac{3}{\sigma^2} \mathbb{E}(Z^2) - \left(\frac{\mathfrak{a} - 1}{\sigma^2}\right)(2b\_{11} - b\_{31} - b\_{22}), \\\\ I\_{\sigma\mu} & = & \frac{1}{\sigma} b\_{11}, \\\\ I\_{\mu\mu} & = & \frac{1}{\mathfrak{a}^2} - \frac{2^\mu \log^2(2)}{(2^\mu - 1)^2}, \end{array}$$

where *bij* = E *Ziκj*(*<sup>Z</sup>*; 0, 1) must be numerically computed.

The Fisher's (expected) information matrix can be obtained by computing the expected values of the above expressions. By taking in this matrix, *α* = 1, we have that *Z* ∼ *<sup>N</sup>*(*μ*, *σ*) and

$$I\_{\mathcal{F}}(\mu,\sigma,a=1) = \begin{pmatrix} \frac{1}{\sigma^2} & 0 & \frac{1}{\sigma}d\_{02} \\ 0 & \frac{2}{\sigma^2} & \frac{1}{\sigma}d\_{12} \\ \frac{1}{\sigma}d\_{02} & \frac{1}{\sigma}d\_{12} & 1-2\log^2(2) \end{pmatrix} \rho$$

where *dij* = ∞−∞ *z<sup>i</sup>φj*(*z*) <sup>1</sup>+<sup>Φ</sup>(*z*) *dz* must be numerically obtained.

The determinant of *IF*(*μ*, *σ*, *α* = 1) is (2 − 4 log<sup>2</sup>(2) − *b*212 − <sup>2</sup>*<sup>b</sup>*202)/*σ*<sup>4</sup> = 0.003357435/*σ*<sup>4</sup> = 0, consequently, the Fisher's information matrix is nonsingular at *α* = 1.

Therefore, for large samples, the ML estimators, *θ* , of *θ* are asymptotically normal, that is,

$$
\sqrt{n}\left(\widehat{\boldsymbol{\theta}}-\boldsymbol{\theta}\right) \stackrel{\mathcal{L}}{\longrightarrow} N\_3(\mathbf{0}, \ I(\boldsymbol{\theta})^{-1}),
$$

resulting in the asymptotic variance of the ML estimators *θ* ˆ being the inverse of Fisher's information matrix *<sup>I</sup>*(*θ*). As the parameters are unknown, the observed information matrix is usually considered, where the unknown parameters are estimated by ML.
