A Generalized-Alpha–Beta-Skew Normal Distribution with Applications

Abstract

Recently there is a lot of research related to skewed distributions and their growing relevance in data analytics. In the present work we introduce a new generalized version of alpha beta skew normal distribution and some of its basic properties are investigated. Some extensions of the proposed distribution have also been studied. A simulation study has been conducted to see the performance of the obtained estimators of the parameters using Metropolis–Hastings (MH) algorithm. The appropriateness of the proposed distribution has been tested by comparing it with twelve closely related and nested distributions using Akaike Information Criterion. The Likelihood Ratio test has been employed for testing the relevance of the induction of the additional parameters in the proposed model.

Appendix

1.1 Derivation of normalizing constant $C(\alpha ,\beta ,\lambda )$

$$\begin{aligned} C(\alpha ,\beta ,\lambda ) & = \int\limits_{ - \infty }^{\infty } {[(1 - \alpha \,z - \beta \,z^{3} )^{2} + 1]\varphi (z)\,\Phi (\lambda \,z)} \,dz \\ & = \int\limits_{ - \infty }^{\infty } {(2 - 2\alpha \,z + \alpha^{2} z^{2} - 2\beta \,z^{3} + 2\alpha \beta \,z^{4} + \beta^{2} z^{6} )\varphi (z)\,\Phi (\lambda \,z)} \,dz \\ & = \int\limits_{ - \infty }^{\infty } {\varphi (z;\lambda )\,} \,dz - \alpha \,E(Z_{\lambda } ) + \frac{{\alpha^{2} }}{2}E(Z_{\lambda }^{2} ) - \beta \,E(Z_{\lambda }^{3} ) + \alpha \beta \,E(Z_{\lambda }^{4} ) + \frac{{\beta^{2} }}{2}\,E(Z_{\lambda }^{6} ) \\ & = 1 + \frac{{\alpha^{2} }}{2} + 3\,\alpha \,\beta + \frac{{15\beta^{2} }}{2} - \alpha \sqrt {\frac{2}{\pi }} \frac{\lambda }{{\sqrt {1 + \lambda^{2} } }} - \beta \sqrt {\frac{2}{\pi }} \frac{\lambda }{{\sqrt {1 + \lambda^{2} } }}\frac{{3 + 2\lambda^{2} }}{{1 + \lambda^{2} }} \\ & = 1 + 3\,\alpha \,\beta - \alpha \,b\delta - \beta \,b\delta \frac{{3 + 2\lambda^{2} }}{{1 + \lambda^{2} }} + \frac{{\alpha^{2} }}{2} + \frac{{15\beta^{2} }}{2}, \\ \end{aligned}$$

where $b = \sqrt {\frac{2}{\pi }} ,\,\,\delta = \frac{\lambda }{{\sqrt {1 + \lambda^{2} } }}$ and $\varphi (z;\lambda )$ is the density function of $Z_{\lambda } \sim SN(\lambda )$.

1.2 Derivation of the cdf

$$\begin{aligned} F_{Z} (z) & = P(Z \le z)\, = \int\limits_{ - \infty }^{z} {\frac{{(1 - \alpha \,t - \beta \,t^{3} )^{2} + 1}}{C(\alpha ,\beta ,\lambda )}} \,\varphi (t)\,\Phi (\lambda t)dt \\ & = \frac{1}{C(\alpha ,\beta ,\lambda )}\int\limits_{ - \infty }^{z} {(2 - 2\alpha \,t + \alpha^{2} t^{2} - 2\beta \,t^{3} + 2\alpha \beta \,t^{4} + \beta^{2} t^{6} )\varphi (t)\,\Phi (\lambda t)dt} \\ & = \frac{1}{C(\alpha ,\beta ,\lambda )}\left[ {\int\limits_{ - \infty }^{z} {2\varphi (t)\,\Phi (\lambda t)dt} \, - 2\alpha \int\limits_{ - \infty }^{z} {t\varphi (t)\,\Phi (\lambda t)dt} \, + \alpha^{2} \int\limits_{ - \infty }^{z} {t^{2} \varphi (t)\,\Phi (\lambda t)dt} \, - 2\beta \int\limits_{ - \infty }^{z} {t^{3} \varphi (t)\,\Phi (\lambda t)dt} \, + } \right. \\ & \left. {\quad 2\alpha \beta \int\limits_{ - \infty }^{z} {t^{4} \varphi (t)\,\Phi (\lambda t)dt} \, + \beta^{2} \int\limits_{ - \infty }^{z} {t^{6} \varphi (t)\,\Phi (\lambda t)dt} } \right] \\ \end{aligned}$$

Putting the above following results we get the desired result.

$$\begin{aligned} \int\limits_{ - \infty }^{z} {2\varphi (t)\,\Phi (\lambda t)dt} & = \Phi (z;\lambda ),\int\limits_{ - \infty }^{z} {t\varphi (t)\,\Phi (\lambda t)dt} = - \varphi (z)\,\Phi (\lambda z) + \frac{{\lambda \,Erf\left( {(z\sqrt {1 + \lambda^{2} } )/\sqrt 2 } \right)}}{{2\sqrt {2\pi } \sqrt {1 + \lambda^{2} } }} \\ \int\limits_{ - \infty }^{z} {t^{2} \varphi (t)\,\Phi (\lambda t)dt} & = - z\varphi (z)\,\Phi (\lambda z) + \frac{{b\,\delta \,\varphi \left( {z\sqrt {1 + \lambda^{2} } } \right)}}{{2\sqrt {1 + \lambda^{2} } }}, \\ \int\limits_{ - \infty }^{z} {t^{3} \varphi (t)\,\Phi (\lambda t)dt} & = - (2 + z^{2} )\varphi (z)\,\Phi (\lambda z) + \frac{{\lambda \,Erf\left( {(z\sqrt {1 + \lambda^{2} } )/\sqrt 2 } \right)}}{{\sqrt {2\pi } \sqrt {1 + \lambda^{2} } }} - z\frac{{b\,\delta \,\varphi \left( {z\sqrt {1 + \lambda^{2} } } \right)}}{{2\sqrt {1 + \lambda^{2} } }} + \frac{{b\,\delta \,\Phi \left( {z\sqrt {1 + \lambda^{2} } } \right)}}{{2\sqrt {1 + \lambda^{2} } }}, \\ \int\limits_{ - \infty }^{z} {t^{4} \varphi (t)\,\Phi (\lambda t)dt} & = - z(3 + z^{2} )\varphi (z)\,\Phi (\lambda z) - \frac{{b\,\delta \,\varphi \left( {z\sqrt {1 + \lambda^{2} } } \right)\,\left( {5 + z^{2} + \lambda^{2} (3 + z^{2} )} \right)}}{{2\left( {\sqrt {1 + \lambda^{2} } } \right)^{3} }} + \frac{3}{2}\Phi (z;\lambda ), \\ \int\limits_{ - \infty }^{z} {t^{6} \varphi (t)\,\Phi (\lambda t)dt} & = - z(15 + 5z^{2} + z^{4} )\varphi (z)\,\Phi (\lambda z) + \frac{15}{2}\Phi (z;\lambda ) + \\ & \quad \frac{{b\,\delta \,\varphi \left( {z\sqrt {1 + \lambda^{2} } } \right)\,\left( { - 33 - 40\lambda^{2} - 15\lambda^{4} - z^{4} (1 + \lambda^{2} )^{2} - z^{2} (9 + 14\lambda^{2} + 5\lambda^{4} )} \right)}}{{2\left( {\sqrt {1 + \lambda^{2} } } \right)^{5} }}. \\ \end{aligned}$$

1.3 Derivation of the moment

$$\begin{aligned} E(Z^{k} ) & = \int\limits_{ - \infty }^{\infty } {z^{k} \frac{{(1 - \alpha z - \beta z^{3} )^{2} + 1}}{C(\alpha ,\beta ,\lambda )}\varphi (z)\,\Phi (\lambda z)\,} dz \\ & = \frac{1}{C(\alpha ,\beta ,\lambda )}\left[ {\int\limits_{ - \infty }^{\infty } {\,2\,z^{k} } \,\varphi (z)\,\Phi (\lambda z)dz - 2\alpha \int\limits_{ - \infty }^{\infty } {\,z^{k + 1} } \varphi (z)\,\Phi (\lambda z)\,dz + \alpha^{2} \int\limits_{ - \infty }^{\infty } {\,z^{k + 2} } \varphi (z)\,\Phi (\lambda z)\,dz - } \right. \\ & \left. {\quad 2\beta \int\limits_{ - \infty }^{\infty } {\,z^{k + 3} } \varphi (z)\,\Phi (\lambda z)\,dz + 2\alpha \beta \int\limits_{ - \infty }^{\infty } {\,z^{k + 4} } \varphi (z)\,\Phi (\lambda z)\,dz + \beta^{2} \int\limits_{ - \infty }^{\infty } {\,z^{k + 6} } \varphi (z)\,\Phi (\lambda z)\,dz} \right] \\ & = \frac{1}{C(\alpha ,\beta ,\lambda )}\left[ {E(Z_{\lambda }^{k} ) - 2\alpha \,E(Z_{\lambda }^{k + 1} ) + \alpha^{2} \,E(Z_{\lambda }^{k + 2} ) - 2\beta \,E(Z_{\lambda }^{k + 3} ) + 2\alpha \beta \,E(Z_{\lambda }^{k + 4} ) + \beta^{2} \,E(Z_{\lambda }^{k + 5} )} \right] \\ \end{aligned}$$