Looking for a distribution where: Mean=0, variance is variable, Skew=0 and kurtosis is variable

Question

I am aiming to run simulations in order to estimate the influence of the distribution of $Y$ (independent variable) on a certain binary outcome $X$ (dependent variable). $Y$ must always has a mean of 0 and must be symmetric (skew = 0) but I'd like to vary independently the variance (eventually from 1 to 250) and the kurtosis (from a square to a narrow peak, eventually from -1.2 to 3) of $Y$.

I've been looking at a variety of distributions but I found none that seemed suited for this kind purpose.

Can you help me choosing a distribution from which I can easily change the variance and kurtosis while keeping the mean and the skew at zero?

EDITS

More descriptions

I realized there were still open questions (not that many though). My preferences are:

• mean = 0
• variance: can vary
• symmetric: Yes (I didn't realized at first that skew=0 doesn't imply symmetry)
• kurtosis: can vary independently of the variance
• number of modes: unimodal (not strict on this issue)
• Bounded: Doesn't matter.
• continuous/discrete: Doesn't matter. I will have to take a discrete approximations if it is continuous but I don't mind
• Tail Behaviour: I am quite interested in the impact of the tail. I thought that by allowing kurtosis to change was enough as a description but maybe I should describe higher moments...

Context

I am working in the field of population genetics and I am interested in the influence of the variance and kurtosis of the dispersal kernel ($Y$) on the probability of fixation of a given allele ($X$). I am not interested to simulate cases where the distribution of dispersal is skewed and the mean should always be at the position of the deme of the parents (this can very easily solved by a simple addition anyway). Previous studies have argued (without showing any evidence) that variance and kurtosis are important and empirical studies showed that the dispersal kernel is often leptokurtic. Not sure I correctly addressed your comment. Did I? Thank you

Answer 1

There are many possibilities. One possibility is to take a pair of families to cover positive and negative excess kurtosis, and when matching the first four moments the obvious candidates are Pearson-family distributions.

The family of scaled t-distributions have parameters ($\sigma$ and $\nu$) that affect the variance and kurtosis. They can only have kurtosis above that of the normal, though.

That will have excess kurtosis $\frac{6}{\nu-4}$ (so if you want that to only go as high as 3, you'll want $\nu\geq 6$).

It has variance $\sigma^2 \frac{\nu}{\nu-2}$, so given $\nu$ you can choose $\sigma$ to yield the desired variance

The family of scaled shifted (to mean 0) beta distributions then would take care of the case where kurtosis was smaller than for the normal (both families include the normal as a limiting case). So take a $\text{Beta}(\alpha,\alpha)$ and shift it down by $\frac{1}{2}$ and then scale to the desired variance.

That is a $\text{Beta}(\alpha,\alpha)$ has excess kurtosis $-\frac{6}{ (2\alpha + 3)}$, and includes your desired uniform at $\alpha=1$.

Before scaling it has variance $\frac{1}{4(2\alpha+1)}$; the ratio of the desired variance to that unscaled variance will be the square of the required scale.

Answer 2

Have a look at heavy tail Lambert W x F distributions (disclaimer: I am the author). The random variable $Y \sim Lambert W \times F$ is a heavy-tail version of $X\sim F$, where you control the tails with tail parameter $\delta \geq 0$: for $\delta = 0$, $X = Y$ and thus Lambert W x F is the same as F; and for $\delta \rightarrow \infty$ you get more and more heavy tails in $Y$ (Tukey's h is a special case of Lambert W x F random variables for F = Gaussian and $\alpha = 1$).

Note that this works for any (non pathological) continuous distribution -- not just the Normal distribution.

For your particular request of mean = 0, symmetric, and variable variance and kurtosis you can set $\mu = 0$, $\delta_{\ell} = \delta_r = \delta$ (by default), and vary scale $\sigma$ and tail parameter $\delta$.

In R this is implemented in the LambertW package (use type = "h" and distname = "normal").

See also (my) related replies here:

• Transformation to increase kurtosis and skewness of normal r.v: this shows some illustrations of how the densities vary when varying $\delta$.
• What's the distribution of these data?: an application example of how to use this to estimate model parameters and Gaussianize your data.

Answer 3

You said you would consider discrete distributions. Perhaps the simplest meeting your conditions is (for some $x>0$ and $0\lt p \le 1$)

• $\mathbb P(X=-x) =\frac p2$
• $\mathbb P(X=0) =1-p$
• $\mathbb P(X=+x) =\frac p2$

which has

• a mean of $\mu=0$,
• a variance of $px^2$,
• a third moment of $0$ so a skewness of $0$, and
• a fourth moment of $px^4$ so a raw kurtosis of $\frac1p$ and an excess kurtosis of $\frac1p -3$.

So you can set

• $p$ to be the reciprocal of the raw kurtosis and
• $x^2$ to be the variance multiplied by the raw kurtosis and take its square root to be $x$

(remembering that the raw kurtosis is $3 +{}$ the excess kurtosis).