# Is there a plateau-shaped distribution?

I am looking for a distribution where the probability density decreases quickly after some point away from the mean, or in my own words a "plateau-shaped distribution".

Something in between the Gaussian and the uniform.

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You could sum a Gaussian RV and a uniform RV. – StrongBad Mar 25 at 14:06
One sometimes hears about so-called platykurtic distributions. – J. M. Mar 26 at 2:10

You may be looking for distribution known under the names of generalized normal (version 1), Subbotin distribution, or exponential power distribution. It is parametrized by location $\mu$, scale $\sigma$ and shape $\beta$ with pdf

$$\frac{\beta}{2\sigma\Gamma(1/\beta)} \exp\left[-\left(\frac{|x-\mu|}{\sigma}\right)^{\beta}\right]$$

as you can notice, for $\beta=1$ it resembles and converges to Laplace distribution, with $\beta=2$ it converges to normal, and when $\beta = \infty$ to uniform distribution.

If you are looking for software that has it implemented, you can check normalp library for R (Mineo and Ruggieri, 2005). What is nice about this package is that, among other things, it implements regression with generalized normally distributed errors, i.e. minimizing $L_p$ norm.

Mineo, A. M., & Ruggieri, M. (2005). A software tool for the exponential power distribution: The normalp package. Journal of Statistical Software, 12(4), 1-24.

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thanks, that is awesome – dontloo Mar 25 at 10:01
Per this answer, the correct term would appear to the "Subbotin distribution." Wikipedia is helpful in many cases, but here it would appear the editors dropped the ball. stats.stackexchange.com/questions/201038/… – General Abrial Mar 25 at 20:54
@user777 that is why I wrote that it is known under the three different names and linked to the same thread ;) – Tim Mar 25 at 23:14
Not sure how I missed that. Sorry! – General Abrial Mar 26 at 1:06

@StrongBad's comment is a really good suggestion. The sum of a uniform RV and gaussian RV can give you exactly what you're looking for if you pick the parameters right. And it actually has a reasonably nice closed form solution.

The pdf of this variable is given by the expression:

$$\dfrac{1}{4a}\left[\mathrm{erf}\left(\dfrac{x+a}{\sigma\sqrt{2}}\right)-\mathrm{erf}\left(\dfrac{x-a}{\sigma\sqrt{2}}\right) \right]$$

$a$ is the "radius" of the zero-mean uniform RV. $\sigma$ is the standard deviation of the zero-mean gaussian RV.

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Reference: Bhattacharjee, G. P., Pandit, S. N. N., and Mohan, R. 1963. Dimensional chains involving rectangular and normal error-distributions. Technometrics, 5, 404–406. – Tim Mar 26 at 11:14

There's an infinite number of "plateau-shaped" distributions.

Were you after something more specific than "in between the Gaussian and the uniform"? That's somewhat vague.

Here's one easy one: you could always stick a half-normal at each end of a uniform:

You can control the "width" of the uniform relative to the scale of the normal so you can have wider or narrower plateaus.

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See my "Devil's tower" distribution in here:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4321753/pdf/nihms-599845.pdf

The "slip-dress"distribution is even more interesting.

It is easy to construct distributions having whatever shape you want.

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