User:Mathstat/Skew elliptical distribution

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In probability theory and statistics, the skew elliptical distribution is a continuous probability distribution that generalizes the family of elliptically symmetric distributions 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
 * $$\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \text{erf} \left(\frac{x}{\sqrt{2}}\right)\right]$$

where erf is the error function. Then the probability density function of the skew-normal distribution with parameter α is given by
 * $$f(x) = 2\phi(x)\Phi(\alpha x). \,$$

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

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, themethod 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) $$|\hat{\gamma}_3| = \min(0.99, |(1/n)\sum{((x_i-\bar{x})/s)^3}|)$$.