A flexible generalization of the skew normal distribution based on a weighted normal distribution

Abstract

The skew normal distribution of Azzalini (Scand J Stat 12:171–178, 1985) has been found suitable for unimodal density but with some skewness present. Through this article, we introduce a flexible extension of the Azzalini (Scand J Stat 12:171–178, 1985) skew normal distribution based on a symmetric component normal distribution (Gui et al. in J Stat Theory Appl 12(1):55–66, 2013). The proposed model can efficiently capture the bimodality, skewness and kurtosis criteria and heavy-tail property. The paper presents various basic properties of this family of distributions and provides two stochastic representations which are useful for obtaining theoretical properties and to simulate from the distribution. Further, maximum likelihood estimation of the parameters is studied numerically by simulation and the distribution is investigated by carrying out comparative fitting of three real datasets.

Appendix 1

1.1 Appendix A.1: score vector and Hessian matrix

Let $x_1 , x_2 ,\ldots , x_n$ is a random sample drawn from the skew symmetric component normal distribution $SSCN(\mu ,\sigma ,\lambda ,\alpha )$, then the log-likelihood function is given by (13). The elements of the score vector are obtained by differentiation

$$\begin{aligned} \ell _{\mu }= & {} \frac{{ - 2\alpha }}{\sigma }\sum \limits _{i = 1}^n {\frac{{z_i }}{{1 + \alpha z_i^2 }}} + \frac{1}{\sigma }\sum \limits _{i = 1}^n {z_i } - \frac{\lambda }{\sigma }\sum \limits _{i = 1}^n {w_i (0)}, \\ \ell _{\sigma }= & {} \frac{{ - 2n}}{\sigma } - \frac{{2\alpha }}{\sigma }\sum \limits _{i = 1}^n {\frac{{z_i^2 }}{{1 + \alpha z_i^2 }}} + \frac{1}{\sigma }\sum \limits _{i = 1}^n {z_i^2 } - \frac{\lambda }{\sigma }\sum \limits _{i = 1}^n {w_i (1)}, \\ \ell _{\lambda }= & {} \sum \limits _{i = 1}^n {w_i (1)} \\ \ell _{\alpha }= & {} \sum \limits _{i = 1}^n {\frac{{z_i^2 }}{{1 + \alpha z_i^2 }}} - \frac{n}{{1 + \alpha }}, \end{aligned}$$

where $z_i = \frac{{x_i - \mu }}{\sigma }$ and $w_i (a) = \sum \limits _{i = 1}^n {z_i^a \frac{{\phi (\lambda z_i )}}{{\varPhi (\lambda z_i )}}}$.

The Hessian matrix, second partial derivatives of the log-likelihood, is given by

$$\begin{aligned} \left( {\begin{array}{*{20}{c}} {{\ell _{\mu ,\mu }}} &{}\quad {{\ell _{\mu ,\sigma }}} &{} \quad {{\ell _{\mu ,\lambda }}} &{}\quad {{\ell _{\mu ,\alpha }}} \\ {{\ell _{\sigma ,\mu }}} &{}\quad {{\ell _{\sigma ,\sigma }}} &{}\quad {{\ell _{\sigma ,\lambda }}} &{}\quad {{\ell _{\sigma ,\alpha }}} \\ {{\ell _{\lambda ,\mu }}} &{}\quad {{\ell _{\lambda ,\sigma }}} &{}\quad {{\ell _{\lambda ,\lambda }}} &{} \quad {{\ell _{\lambda ,\alpha }}} \\ {{\ell _{\alpha ,\mu }}} &{}\quad {{\ell _{\alpha ,\sigma }}} &{}\quad {{\ell _{\alpha ,\lambda }}} &{}\quad {{\ell _{\alpha ,\alpha }}} \\ \end{array}} \right) \end{aligned}$$

where

$$\begin{aligned} \ell _{\mu ,\mu }= & {} \frac{1}{{\sigma ^2 }}\left\{ {2\alpha \sum \limits _{i = 1}^n {\frac{{1 - \alpha z_i^2 }}{{(1 + \alpha z_i^2 )^2 }}} - n - \lambda ^2 \sum \limits _{i = 1}^n {\frac{{\phi (\lambda z_i )\left( {\lambda z_i \varPhi (\lambda z_i ) + \phi (\lambda z_i )} \right) }}{{\varPhi ^2 (\lambda z_i )}}} } \right\} ,\\ \ell _{\mu ,\sigma }= & {} \ell _{\sigma ,\mu } = \frac{1}{{\sigma ^2 }}\left\{ {4\alpha \sum \limits _{i = 1}^n {\frac{{z_i + \alpha z_i^3 - \alpha z_i^4 }}{{(1 + \alpha z_i^2 )^2 }}} \, - 1\sum \limits _{i = 1}^n {z_i } } \right. \\&\qquad \left. {\,+ \lambda \sum \limits _{i = 1}^n {\frac{{\phi (\lambda z_i )\left( {\varPhi (\lambda z_i ) - \lambda ^2 z_i^2 \varPhi (\lambda z_i ) - \lambda z_i \phi (\lambda z_i )} \right) }}{{\varPhi ^2 (\lambda z_i )}}} } \right\} , \\ \ell _{\mu ,\alpha }= & {} \ell _{\alpha ,\mu } = \frac{{ - 2}}{\sigma }\sum \limits _{i = 1}^n {\frac{{z_i }}{{(1 + \alpha z_i^2 )^2 }}}, \\ \ell _{\mu ,\lambda }= & {} \ell _{\lambda ,\mu } = \frac{{ - 1}}{\sigma }\sum \limits _{i = 1}^n {\frac{{\phi (\lambda z_i )\left( {\varPhi (\lambda z_i ) - \lambda ^2 z_i^2 \varPhi (\lambda z_i ) - \lambda z_i \phi (\lambda z_i )} \right) }}{{\varPhi ^2 (\lambda z_i )}}}, \\ \ell _{\sigma ,\sigma }= & {} \frac{1}{{\sigma ^2 }}\left\{ {2n + 2\alpha \sum \limits _{i = 1}^n {\frac{{\alpha z_i^4 - z_i^2 }}{{(1 + \alpha z_i^2 )^2 }} - 3\sum \limits _{i = 1}^n {z_i^2 } } } \right. \\&\left. \qquad {\,+ \lambda \sum \limits _{\iota = 1}^n {\frac{{z_i \phi (\lambda z_i )\left( {\varPhi (\lambda z_i ) - (\lambda ^2 z_i^2 - 1)\varPhi (\lambda z_i ) + \lambda z_i \phi (\lambda z_i )} \right) }}{{\varPhi ^2 (\lambda z_i )}}} } \right\} , \\ \ell _{\sigma ,\alpha }= & {} \ell _{\alpha ,\sigma } = \frac{{ - 2}}{\sigma }\sum \limits _{i = 1}^n {\frac{{z_i^2 }}{{(1 + \alpha z_i^2 )^2 }}},\\ \ell _{\sigma \lambda }= & {} \ell _{\lambda \sigma } = \frac{1}{\sigma }\sum \limits _{i = 1}^n {\frac{{z_i \phi (\lambda z_i )\left( { - \varPhi (\lambda z_i ) + \lambda ^2 z_i^2 \varPhi (\lambda z_i ) + \lambda z_i \phi (\lambda z_i )} \right) }}{{\varPhi ^2 (\lambda z_i )}}},\\ \ell _{\alpha ,\alpha }= & {} \frac{n}{{(1 + \alpha )^2 }} - \sum \limits _{i = 1}^n {\frac{{z_i^4 }}{{\left( {1 + \alpha z_i^2 } \right) }}},\\ \ell _{\lambda \alpha }= & {} \ell _{\alpha \lambda } = 0, \\ \ell _{\lambda \lambda }= & {} \sum \limits _{i = 1}^n {\frac{{z_i^2 \left( { - \lambda z_i \phi (\lambda z_i )\varPhi (\lambda z_i ) - \phi ^2 (\lambda z_i )} \right) }}{{\varPhi ^2 (\lambda z_i )}}}. \end{aligned}$$