# What is a 'log-skew normal distribution'?

Recently, I read in a paper that they used a "log-skew normal distribution" to model returns on trades.

I don't have a formal statistics background and was not aware of this distribution type. I am only familiar with the log-normal and skew normal distributions. I assume that a log-skew normal distribution is a variant featuring both?

So, what exactly is a "log-skew normal distribution"? Is it a skewed normal distribution that is log transformed?

• +1 Welcome to CV, John Doe! Your question inspired me to ask Is there a generalized concept of noncentrality of a distribution?. I think your question might be improved if you edit (link at lower left) to add a citation to the paper you mention in your first sentence. Apr 26, 2021 at 20:02
• Most such terminology is borrowed from that of the lognormal distribution. It's the opposite of what you might think: $Z$ has a lognormal distribution exactly when $\log(Z)$ has a normal distribution. That is, $Z$ is the exponential ("antilog") of a normal variate. Similarly, one would expect a variable with a "log skew-normal distribution" to be the exponential of a skew-normal variable. But not all authors get that right: the context matters. However, since many skew-normal variates can take on negative values, it's unlikely anyone would take the logarithm of one.
– whuber
Apr 26, 2021 at 20:06
• If you could link the paper, it could be helpful. A priori you could use a variable whose log is skew-normal to model prices but not returns, so either your description is incorrect or that's not what they mean by "log-skew normal". Apr 27, 2021 at 0:59
• stats.stackexchange.com/questions/486053/… Apr 27, 2021 at 14:47

If $$X$$ has a normal distribution, then $$e^X$$ has a log-normal distribution. That might seem backwards (and the terminology is not used consistently), but the logic is that the logarithm of a log-normal variable is normal. So read the name in reverse, as normal after taking logarithm. The same idea can be used with other distributions on the positive real line, and, in particular, if the logarithm of the variable is skew-normal, we say the variable is log-skew-normal. This paper proposes this as a distribution for insurance claims. As pointed out in a comment, this cannot be used as a model for returns, which unfortunately can be negative ... This arXiv paper uses the log-skew-normal to fit the distribution of sums of independent lognormals.