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

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?

• 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.
– whuber
Jul 6, 2021 at 22:05
• @whuber Oh, too bad. Thanks for updating me. Jul 6, 2021 at 22:12
• Westfall 2014 does a sort of 'debunking paper' on the topic. Jul 6, 2021 at 22:36
• 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. Jul 6, 2021 at 22:36
• Thanks @BruceET , I am ahead of you on that suggestion but it is appreciated. Jul 6, 2021 at 22:37