# Finding a distribution where skewness and kurtosis do not depend on each other. Does it even make sense?

I am simulating non-normal data to investigate how this affects some diagnostical methods that assume normality. In particular I'm interested in seeing how skewness and kurtosis affects the results.

I've investigated skewness using the skewed normal distribution as well as the gamma distribution, but would like to change kurtosis on top of that. Unfortunately this doesn't work for these two distributions as the skewness and kurtosis depend on the same parameter.

The best I've found so far was the skewed student t-distribution in the package "skewt" in R. https://cran.r-project.org/web/packages/skewt/skewt.pdf This seems to, on the surface, do what I would expect: given a constant skew parameter, the visible skewness does not change (the peak remains in place) when I alter degrees of freedom. It presents its own problems however, because I am yet to be able to compute kurtosis for a given value of gamma, and negative kurtosis is impossible to implement.

Are there any other distributions I could use? Does what I ask even make sense? Part of me thinks it does not given the definitions of kurtosis and skewness in terms of moments, but it also seems like it should be doable to maintain a given skew while only "lifting" or "sinking" the tails, as it were.

• Abstractly there is a space of distributions. If, within that space, you plot a curve, then along the curve the skewness and kurtosis obviously depend on each other. But if you consider any region of two or more dimensions that doesn't lie along a level set of either skewness or kurtosis, then you have a family of distributions satisfying your requirements. I hope that helps make it clear that your search for a suitable family of distributions must be guided by more than this weak criterion. Focus instead on distributions likely to be appropriate for you application.
– whuber
Commented Dec 17, 2023 at 17:01
• Discrete distributions are easy to play with and offer great flexibility. And if you must have continuous distributions, you can just add noise. See .stats.stackexchange.com/a/483215/102879 Commented Dec 18, 2023 at 0:37

Finding a distribution where skewness and kurtosis do not depend on each other.

Presumably you mean the standardized third and fourth moments (rather than some of the other measures of skewness and kurtosis).

This is not really possible to do as a general thing because kurtosis is always at least skewness$$^2 + 1$$. Which means, for example, you simply cannot find any distribution that has, say skewness $$3$$ that has kurtosis less than $$10$$ (excess kurtosis less than 7); they're not possible to separate in that sense.

But within the possible region, there are collections of distributions that cover the whole range of possibilities. Perhaps the best known are the Pearson family of distributions. There are many posts about them on the site.

https://en.wikipedia.org/wiki/Pearson_distribution

The Pearson plot (a diagram plotting the regions covered by the family members on the axes of skewness vs kurtosis or skewness$$^2$$ vs kurtosis) is a convenient way to find the distribution in the family that includes a member with the selected skewness and kurtosis.

There are other families that people use that might suit you better, perhaps, such as the Johnson distributions including the $$S_U$$ and $$S_B$$ distributions:

https://en.wikipedia.org/wiki/Johnson%27s_SU-distribution

https://en.wikipedia.org/wiki/Johnson%27s_SU-distribution#Johnson's_SB-distribution

There have been attempts to make distributions that let you tweak the parameters "individually" (at least sort of), but they cannot avoid the fundamental restriction that once you fix skewness, kurtosis has a hard lower bound that's higher than what it would be if skewness were $$0$$. Equivalently, if you choose kurtosis, there are limits on how big skewness can be.

One example of an attempt to separate out the skewness and kurtosis dials is Tukey's $$g$$-and-$$h$$ distribution family, which rely on transformations of normal random variables.

There's some related transformation-based suggestions here

I've seen at least a couple of other such attempts at individual control, but I can't say I found any of them particularly satisfactory for the sort of application you're after; there always seem to be a few less than ideal aspects. I'll try to locate another one of these that I have in mind.

You could use a distribution on just three points to illustrate arbitrary skewness and kurtosis, so long as the raw kurtosis is at least $$1+{}$$ the square of the skewness. Even this gives you more freedom than you need, so you can put one of the points at the mean.

Suppose the skewness is $$\gamma$$ and the raw kurtosis is $$\kappa$$. The following distribution will give this with arbitrary mean $$\mu$$ and variance $$\sigma^2$$, though you can set these equal to $$0$$ and $$1$$ if you wish.

$$\begin{array}{lcl} \mathbb P \Big(X = \mu+\frac12\left(\gamma+\sqrt{4\kappa-3\gamma^2}\right)\sigma\Big) &=& \dfrac{\sqrt{4\kappa-3\gamma^2}-\gamma}{2\sqrt{4\kappa-3\gamma^2}(\kappa-\gamma^2)} \\ \mathbb P \Big(X = \mu\Big) &=& 1-\dfrac{1}{(\kappa-\gamma^2)} \\ \mathbb P\Big(X = \mu+\frac12\left(\gamma-\sqrt{4\kappa-3\gamma^2}\right)\sigma\Big) &=& \dfrac{\sqrt{4\kappa-3\gamma^2}+\gamma}{2\sqrt{4\kappa-3\gamma^2}(\kappa-\gamma^2)}\end{array}$$

These locations and probabilities are continuous functions of the parameters, so you can make a small change to one of skewness or kurtosis without changing the other to see the effect on the locations and probabilities of the three points.

• This would be easier to read using $u=\sqrt{4\kappa-3\gamma^2}$, and writing the formulas in terms of $u$.
– user225256
Commented Dec 19, 2023 at 5:10
• @MattF. Something like: $\mathbb P \Big(X = \mu+\frac12\left(\gamma+u\right)\sigma\Big) = \dfrac{u-\gamma}{2u(\kappa-\gamma^2)}$, $\mathbb P \Big(X = \mu\Big) = 1-\dfrac{1}{(\kappa-\gamma^2)}$, $\mathbb P\Big(X = \mu+\frac12\left(\gamma-u\right)\sigma\Big) = \dfrac{u+\gamma}{2u(\kappa-\gamma^2)}$ using $u=\sqrt{4\kappa-3\gamma^2}$ Commented Dec 19, 2023 at 9:01