Table 1.

Summary of the asymmetric distributions skew-normal (SN), power-normal (PN), and alpha-power skew-normal (SNAP)

Distribution	PDF	CDF	Mean	Median	Skewness	Kurtosis
$SN(\lambda )$	$2\varphi (z)\mathrm{\Phi }(\lambda z)$	$\mathrm{\Phi }(z)-2T(z;\lambda )$	$\mu +\sqrt{\frac{2}{\pi }}\sigma \frac{\lambda }{\sqrt{1+{\lambda }^2}}$	$\mu +\sigma {\mathrm{\Phi }}_{\mathit{SN}}^{-1}(0.5;\lambda )$	$[-0.9953,0.9953]$	[3, 3.8692]
$PN(\alpha )$	$\alpha \varphi (z){\{\mathrm{\Phi }(z)\}}^{\alpha -1}$	${\{\mathrm{\Phi }(z)\}}^{\alpha }$	$\mu +\alpha \sigma {\int }_0^1{\mathrm{\Phi }}^{-1}(u)u^{\alpha -1}du$	$\mu +\sigma {\mathrm{\Phi }}^{-1}(0.5^{1/\alpha })$	$[-0.6115,0.9007]$	[1.7170, 4.3556]
$SNAP(\lambda ,\alpha )$	$\alpha {\varphi }_{\mathit{SN}}(z;\lambda ){\{{\mathrm{\Phi }}_{\mathit{SN}}(z;\lambda )\}}^{\alpha -1}$	${\{\mathrm{\Phi }(z)-2T(z;\lambda )\}}^{\alpha }$	$\mu +\alpha \sigma {\int }_0^1{\mathrm{\Phi }}_{\mathit{SN}}^{-1}(u;\lambda )u^{\alpha -1}du$	$\mu +\sigma {\mathrm{\Phi }}_{\mathit{SN}}^{-1}(0.5^{1/\alpha };\lambda )$	$[-1.4676,0.9953]$	[1.4672, 5.4386]

$\varphi (·)$ and $\mathrm{\Phi }(·)$ denote the PDF and CDF of the Normal distribution; ${\varphi }_{\mathit{SN}}(·)$ and ${\mathrm{\Phi }}_{\mathit{SN}}(·;\lambda )$ are the PDF and CDF of the standard Skew-Normal distribution; ${\mathrm{\Phi }}^{-1}(·)$ and ${\mathrm{\Phi }}_{\mathit{SN}}^{-1}(·;\lambda )$ denote the inverse functions of ${\varphi }_{\mathit{SN}}(·)$ and ${\mathrm{\Phi }}_{\mathit{SN}}(·;\lambda )$ respectively, and $T(z;\lambda )$ is the Owen’s T function $z=\frac{x-\mu }{\sigma }$. As skewness and kurtosis do not have closed-form expressions, these are estimated numerically by integrating over the first, second, third, and fourth moments