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(This comes from a Facebook post, reference at the end)

"Everybody knows" that the function $x \mapsto \exp\left(-1/x^2 \right)$ is infinitely differentiable, but not (real) analytic, since all its derivatives at zero are 0, so its MacLaurin series is identically zero. That function is very flat at the origin. If we flip it horizontally we get a density function, which I will call the Altiplano distribution: $$ f(x)= \frac{1 - e^{-1/x^2}}{2\sqrt{\pi}} $$ This can be extended to a location-scale-family in the usual way:

Altiplano distribution example densities

This distribution does not have an expectation, in fact its tails are close to Cauchy tails. In fact, a Laurent series expansion (at infinity) shows that the tails behave (to first order) as $x^{-2}$.

Other interesting examples of distributions very flat around the center? Some earlier mentions of this specific distribution?

Reference for this example

Facebook post by Dr. Arturo Erdély of UNAM. People with a facebook account can probably find it.

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    $\begingroup$ Interesting (+1)! I could see this being useful as a low-information prior on a location parameter. $\endgroup$
    – jbowman
    Commented May 25 at 22:50
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    $\begingroup$ +1 Kjetil for posting this as I couldn't access the fb link (for not having an account maybe?). $\endgroup$ Commented May 26 at 2:25
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    $\begingroup$ @jbowman Thank you for pointing out that application. I may use it. $\endgroup$
    – Galen
    Commented May 26 at 3:27
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    $\begingroup$ Another example: stats.stackexchange.com/q/201038/22311 $\endgroup$
    – Sycorax
    Commented May 26 at 3:34
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    $\begingroup$ @NickCox I've used wrapped distributions in modeling turning probabilities for animal movement. I haven't looked into them before, but I could see a flat-top wrapped distribution making sense for this as well. Possibly with the the width of the top being correlated to something like the field of view an animal has without turning their body. $\endgroup$
    – anjama
    Commented May 27 at 0:52

2 Answers 2

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Other interesting examples of distributions very flat around the center?

I find the generalized normal distribution interesting. Its PDF is given by

$$\frac{\beta}{2 \alpha \Gamma \left(\frac{1}{\beta} \right)}e^{- \left( \frac{| x-\mu |}{\alpha} \right)^{\beta}}.$$

It gives a normal distribution when $\beta=2$ and $\sigma^2 = \frac{\alpha^2}{2}$ and gives a Laplace distribution when $\beta = 1$.

The flatness around the mode depends upon its $\beta$ parameter, and a pointwise convergence to a uniform density occurs on the interval $(\mu - \alpha, \mu + \alpha)$ as $\beta \rightarrow \infty$. So the parameter $\beta$ gives you how flat the distribution should be near the mode, and the size of the scale $\alpha$ tells you how close counts as "near the mode".

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    $\begingroup$ It also goes by the name 'exponential power' distribution. $\endgroup$ Commented May 26 at 5:14
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The mixture distribution of $Z=X+Y$ where $X$ is a standard normal and independently $Y=\pm1$ with equal probability, with mean/median/mode of $0$ and variance of $2$, has a flat-topped density $$\frac{1}{\sqrt{8\pi}}\left(e^{-(x-1)^2/2} + e^{-(x+1)^2/2}\right)$$ (much flatter than the corresponding normal distribution) looking like

enter image description here

As @MichaelLugo commented, you can look at the Taylor expansion of the density around $0$ to get $$\sqrt{\frac{1}{2\pi e}}-\sqrt{\frac{1}{288\pi e}}x^4 + O(x^6)$$ and the lack of an $x^2$ term explains the flatness near $x=0$.

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  • $\begingroup$ Two things: (1). The exponents should be negative. (2) If you expand that density in a power series around $x = 0$ there's no $x^2$ term, which explains the flatness. $\endgroup$ Commented May 28 at 13:48
  • $\begingroup$ @MichaelLugo Yes - thank you - edited $\endgroup$
    – Henry
    Commented May 28 at 14:12

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