# Estimate parameters for skew normal distribution

I know there is already at least one question answer with this, but I think the solution does not apply in my case.

I have a population for which I know the mean, variance and skewness. I saw how to estimate the parameters for the skew normal distribution given this data, but the problem is that when calculating $\alpha$, $\delta > 1$ (by wikipedia notation), and thus $\alpha$ is undetermined?

How can I solve this?

By looking at the formulae in the wikipedia page, one can see that if we know skewness, we know $\delta$ (note that skewness is monotonic in $\delta$, so the relation is unique). In turn this determines the sign of $\alpha$.
Also, $$|\delta| = \frac{|\alpha|}{\sqrt{1+\alpha^2}} \implies \delta^2(1+\alpha^2)=\alpha^2 \implies (\delta^2-1)\alpha^2 +\delta^2 = 0$$
So if $\delta \geq1$ this is impossible, not undetermined.
And indeed, this is one of the limitations of the skew normal distribution: $\delta$ should be strictly smaller than unity, which in turn imposes limits on the magnitude of the skewness coefficient
So if someone specifies a skew normal distribution with such a theoretical skew coefficient, that $\delta$ ends up equal or above unity, it is simply wrong.
When the sample skew coefficient is higher than theoretically possible, then one can apply maximum likelihood to estimate $\alpha$.