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Skewness is generally defined as a standardized third-order centered moment.

$$S(X) \triangleq \frac{\mathbb{E}\left[ \left( X - \mathbb{E} \left[ X \right] \right)^3 \right]}{\left( \mathbb{E}\left[ \left( X - \mathbb{E} \left[ X \right] \right)^2 \right] \right)^{\frac{3}{2}}}$$

Sometimes skewness is the property that we're interested in, but often it is used to quantify the reflective asymmetry of a probability distribution. This is a heuristic because Meijer 2000 demonstrated that asymmetric distributions can have a skewness of zero even though a non-zero skewness implies that a distribution is asymmetric.

Drawing from skewness as an analogy, I would like to consider kurtosis under a similar light. Kurtosis is likewise a high-order moment:

$$K(X) \triangleq \frac{\mathbb{E}\left[ \left( X - \mathbb{E} \left[ X \right] \right)^4 \right]}{\left( \mathbb{E}\left[ \left( X - \mathbb{E} \left[ X \right] \right)^2 \right] \right)^{2}}$$

Analogously, as skewness tells us about reflective symmetry, kurtosis tells us about how 'tall-and-skinny' the distribution is. As whuber discusses on another post, the symmetry property can be formulated as an equality in terms of either its CDF or PDF (assuming sufficient smoothness).

I have yet to see a formulation of the 'tall-and-skinny' property of a distribution written in an analogous way in terms of either the CDF or PDF of a distribution.

Off the top of my head I would speculate that kurtosis of a CDF could involve a comparison to either extreme of a step function being a maximally-kurtic peak, and a line being a minimally-kurtic peak.

In terms of the PDF I might suspect that an aspect-ratio could be suitable, except that some choice on what interval counts as the 'base' would need to be defined. The choice of height might be taken to be the density at the maximum likelihood value (MLV), but this would be problematic for distributions with multiple MLV's.

How can a 'tall-and-skinny' property related to kurtosis be stated rigorously in terms of its (smooth) probability functions?

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    $\begingroup$ The meaning and interpretation of kurtosis has become somewhat of a controversy here on CV. See this site search. The problem is that kurtosis isn't actually about the height of a density function or its tail behavior, because the fourth moment doesn't directly relate to either. $\endgroup$
    – whuber ♦
    Jul 6, 2021 at 22:05
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    $\begingroup$ @whuber Oh, too bad. Thanks for updating me. $\endgroup$
    – Galen
    Jul 6, 2021 at 22:12
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    $\begingroup$ Westfall 2014 does a sort of 'debunking paper' on the topic. $\endgroup$
    – Galen
    Jul 6, 2021 at 22:36
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    $\begingroup$ For various reasons it is difficult to get an intuitive idea of kurtosis. I think that looking at examples might be the best way to start. This Wikipedia page might be helpful. $\endgroup$
    – BruceET
    Jul 6, 2021 at 22:36
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    $\begingroup$ Thanks @BruceET , I am ahead of you on that suggestion but it is appreciated. $\endgroup$
    – Galen
    Jul 6, 2021 at 22:37

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Well, you cannot relate tall and skinny to kurtosis because there is no mathematical connection. The beta(.5,1) distribution is infinitely tall and skinny but has low kurtosis. And the .9999U(0,1) + .0001Cauchy mixture appears perfectly flat over 99.99% of the observable data, but has infinite kurtosis.

Contrary to whuber's comment, kurtosis is in fact precisely related to the tails. Larger kurtosis mathematically implies greater tail leverage. See here and here for precise descriptions of "tail leverage."

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  • $\begingroup$ Interesting. Does this lack of relation include a lack of stochastic dependence? $\endgroup$
    – Galen
    Jul 7, 2021 at 3:16
  • $\begingroup$ I recall from earlier today that Westfall 2014 provides a mathematical argument for tails being more influential than the central values on the value of kurtosis of a distribution. $\endgroup$
    – Galen
    Jul 7, 2021 at 3:17
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    $\begingroup$ I like the counterexamples you give that illustrate the kurtosis cannot be a direct or monotonic function of what I intuitively recognise as skinny-and-tallness. $\endgroup$
    – Galen
    Jul 7, 2021 at 3:24
  • $\begingroup$ Right. But there is a monotonic relation between kurtosis and tail leverage, as my linked post shows. $\endgroup$
    – BigBendRegion
    Jul 7, 2021 at 8:56
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    $\begingroup$ There would have to be many more details, but generally probably not. For example, the sample mean and sample variance are correlated under non-normality. $\endgroup$
    – BigBendRegion
    Jul 7, 2021 at 14:23

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