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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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6  
You could sum a Gaussian RV and a uniform RV. – StrongBad Mar 25 at 14:06
1  
One sometimes hears about so-called platykurtic distributions. – J. M. Mar 26 at 2:10
up vote 40 down vote accepted

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.

enter image description here

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
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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.

PDFs

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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:

enter image description here

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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