Skew normal distribution

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Parameters Probability density function Cumulative distribution function $\xi \,$ location (real) $\omega \,$ scale (positive, real) $\alpha \,$ shape (real) $x \in (-\infty; +\infty)\!$ $\frac{1}{\omega\pi} e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)} e^{-\frac{t^2}{2}}\ dt$ $\Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right)$ $T(h,a)$ is Owen's T function $\xi + \omega\delta\sqrt{\frac{2}{\pi}}$ where $\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}$ $\omega^2\left(1 - \frac{2\delta^2}{\pi}\right)$ $\gamma_1 = \frac{4-\pi}{2} \frac{\left(\delta\sqrt{2/\pi}\right)^3}{ \left(1-2\delta^2/\pi\right)^{3/2}}$ $2(\pi - 3)\frac{\left(\delta\sqrt{2/\pi}\right)^4}{\left(1-2\delta^2/\pi\right)^2}$ $M_X\left(t\right)=2\exp\left(\xi t+\frac{\omega^2t^2}{2}\right)\Phi\left(\omega\delta t\right)$

In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

Definition

Let $\phi(x)$ denote the standard normal probability density function

$\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$

with the cumulative distribution function given by

$\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right]$,

where erf is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter $\alpha$ is given by

$f(x) = 2\phi(x)\Phi(\alpha x). \,$

This distribution was first introduced by O'Hagan and Leonard (1976).

A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara (1984). Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong (1986), which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others.

To add location and scale parameters to this, one makes the usual transform $x\rightarrow\frac{x-\xi}{\omega}$. One can verify that the normal distribution is recovered when $\alpha = 0$, and that the absolute value of the skewness increases as the absolute value of $\alpha$ increases. The distribution is right skewed if $\alpha>0$ and is left skewed if $\alpha<0$. The probability density function with location $\xi$, scale $\omega$, and parameter $\alpha$ becomes

$f(x) = \frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha \left(\frac{x-\xi}{\omega}\right)\right). \,$

Note, however, that the skewness of the distribution is limited to the interval $(-1,1)$.

Estimation

Maximum likelihood estimates for $\xi$, $\omega$, and $\alpha$ can be computed numerically, but no closed-form expression for the estimates is available unless $\alpha=0$. If a closed-form expression is needed, the method of moments can be applied to estimate $\alpha$ from the sample skew, by inverting the skewness equation. This yields the estimate

$|\delta| = \sqrt{\frac{\pi}{2} \frac{ |\hat{\gamma}_3|^{\frac{2}{3}} }{|\hat{\gamma}_3|^{\frac{2}{3}}+((4-\pi)/2)^\frac{2}{3}}}$

where $\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}$, and $\hat{\gamma}_3$ is the sample skew. The sign of $\delta$ is the same as the sign of $\hat{\gamma}_3$. Consequently, $\hat{\alpha} = \delta/\sqrt{1-\delta^2}$.

The maximum (theoretical) skewness is obtained by setting ${\delta = 1}$ in the skewness equation, giving $\gamma_3 \approx 0.9952717$. However it is possible that the sample skewness is larger, and then $\alpha$ cannot be determined from these equations. When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example) $|\hat{\gamma}_3| = \min(0.99, |(1/n)\sum{((x_i-\bar{x})/s)^3}|)$.

Differential equation

The differential equation leading to the pdf of the skew normal distribution is

$\omega^4 f''(x)+\left(\alpha^2+2\right) \omega^2 (x-\zeta) f'(x)+f(x) \left(\left(\alpha^2+1\right) (x-\zeta )^2+\omega^2\right)=0$,

with initial conditions

$\begin{array}{l} \displaystyle f(0)=\frac{\exp\left(-\frac{\zeta^2}{2\omega^2}\right) \operatorname{erfc}\left(\frac{\alpha\zeta}{\sqrt{2} \omega}\right)} {\sqrt{2\pi}\omega} \text{ and} \\[16pt] \displaystyle f'(0)=\frac{\exp\left(-\frac{\left(\alpha^2+1\right)\zeta ^2} {2 \omega^2}\right) \left(2\alpha\omega+\sqrt{2\pi} \zeta \exp\left(\frac{\alpha^2 \zeta^2}{2 \omega^2}\right) \operatorname{erfc}\left(\frac{\alpha\zeta}{\sqrt{2} \omega}\right)\right)} {2\pi\omega^3}. \end{array}$

References

• Andel, J., Netuka, I. and Zvara, K. (1984). On threshold autoregressive processes. Kybernetika, 20, 89-106.
• Azzalini, A. (1985). "A class of distributions which includes the normal ones". Scandinavian Journal of Statistics 12: 171–178.
• Chan, K-S. and Tong, H. (1986). A note on certain integral equations associated with non-linear time series analysis. Probability and Related Fields, 73, 153-158.
• O'Hagan, A. and Leonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints. Biometrika, 63, 201-202.