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According to Wikipedia, the only correct interpretation of kurtosis is "tail extremity," the logic being that datapoints within one standard deviation of the mean are raised to the fourth power and don't contribute to the expected value $\mathbb{E}(\frac{X-\mu}{\sigma})^4$ since they are much less than 1.

As Westfall (2014) notes, "...its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution)." The logic is simple: Kurtosis is the average (or expected value) of the standardized data raised to the fourth power. Any standardized values that are less than 1 (i.e., data within one standard deviation of the mean, where the "peak" would be), contribute virtually nothing to kurtosis, since raising a number that is less than 1 to the fourth power makes it closer to zero. The only data values (observed or observable) that contribute to kurtosis in any meaningful way are those outside the region of the peak; i.e., the outliers. Therefore, kurtosis measures outliers only; it measures nothing about the "peak".

If that's the reasoning, wouldn't higher moments be a better measure of tailed-ness? E.g. if we take $\mathbb{E}(\frac{X-\mu}{\sigma})^6$, the points within one standard deviation of the mean would contribute even less and points outside would contribute even more. Is there any formal reasoning for why the 4th power is the "correct" one, and not the 6th, 10th, 1000th?

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    $\begingroup$ The text claims that kurtosis captures tail extremity, not that tail extremity is only captured by kurtosis. $\endgroup$ – mkt - Reinstate Monica Sep 29 '19 at 18:11
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There are infinitely many measures of tail heaviness; kurtosis is just one of them. Certainly, higher moments also measure tail extremity, only with emphasis on points farther out in the tail. A limitation of all these is finite moments.

There are also the definitions of heavy-tailedness given on the Wikipedia page https://en.wikipedia.org/wiki/Heavy-tailed_distribution, and these do not require finite moments, but they are limited in the sense that they require the distribution to be unbounded. This is a limitation because many real processes are both bounded and prone to extreme observations.

There does not appear to be any consensus as to what regards "heavy tails," other than that it implies that potentially observable data will occasionally be very far from the main body.

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