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I want to understand the difference between these distribution types.

What is the difference between a long and short-tailed distribution?

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    $\begingroup$ en.wikipedia.org/wiki/Long_tail $\endgroup$ – whuber Jun 12 '17 at 21:33
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    $\begingroup$ @Melih different writers can sometimes mean different things when they say long tail or short tail (for some long means heavier than Gaussian, for some it means heavier than exponential, for some it means asymptotically power-law, and so on). Can you give some context which might narrow down what definition might be intended? $\endgroup$ – Glen_b Jun 13 '17 at 1:18
  • $\begingroup$ We propose a new model instead of the Gaussian graphical model in our study and the distribution of this model is long-tail symmetric since it covers the Gaussian distribution. The professor at the committee asked why we did not use the short-tailed distribution and I do not have much knowledge on this topic, therefore I asked. Thank you for your interest @Glen_b. $\endgroup$ – Melih Aras Jun 13 '17 at 7:13
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    $\begingroup$ That information seems to be directly relevant to potential answers. What was the tail of the proposed distribution like? In what sense do you mean "covers" there? (Did the professor explain why he or she thought the short tail vs long tail issue was important?) $\endgroup$ – Glen_b Jun 13 '17 at 7:15
  • $\begingroup$ I mean, heavier than Gaussian. No he didn't explain, he just asked me to compare their differences. $\endgroup$ – Melih Aras Jun 13 '17 at 7:17
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The distinction which is usually made is between heavy tailed distributions and distributions where the tails decay exponentially (short-tailed distributions).

The tails of these short tail distributions fall off very quickly, while longer-tailed distributions do not. The tails of distributions with "short tails" look like $e^{-x}$.

As a visual example, look at the following, which illustrates the different shape of a cauchy vs normal distribution. Note that much more of the mass of the cauchy distribution is in the tails, whereas most of the mass of the normal distribution is in the center of the distribution.

enter image description here

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

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    $\begingroup$ (-1) 1. "Exponential family" does not refer to exponential tail decay (check your reference [1]). 2. Note that you cannot say that, say, normal distributions and exponential distributions belong "to the exponential family". Instead, the normal distributions are an exponential family, the exponential distributions are an(other) exponential family etc. 3. For example, Pareto distributions with fixed minimum are an exponential family, for example with minimum $1$: $p(x ;\alpha) = \alpha\,x^{-(\alpha+1)} = (\alpha) \, (1) \, \exp((-(a+1))\, (\ln x))$. However, they are heavy-tailed, too. $\endgroup$ – Juho Kokkala Jun 13 '17 at 17:08
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    $\begingroup$ @JuhoKokkala - Thanks for your informative comment! I didn't realize I was misusing that term, it appears I've been conflating "exponential family" and "exponential tail decay" without realizing I was doing so. $\endgroup$ – Louis Cialdella Jun 13 '17 at 22:20
  • $\begingroup$ Your answer is confusing, because you mix up distributions with finite tails (Binomial), exponential tails (Exponential), and rapidly-decreasing tails (Normal). It is difficult, if not impossible, to sort out what you are trying to say about them. Could you clear this up? $\endgroup$ – whuber Oct 31 '18 at 20:38

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