Wikipedia says kurtosis only measures tailedness but not peakedness. But I remember my teacher said several times that high excess kurtosis usually corresponds to fat tails AND thin peak. High excess kurtosis accompanied by fat tails can be easily seen by the usual definition of kurtosis(fourth central moment). But what about peakedness? If kurtosis doesn't measure it, is there any statistic that can do the job? My Statistics textbook isn't clear about this part.


From Kurtosis definition:

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."

In the past it was believed that it measured also the peak of the distribution, which has come to be false.


To elaborate on kubox's correct assertion that kurtosis measures nothing about the peak, it is also thought that perhaps kurtosis measures probability concentration inside the $\mu \pm \sigma$ range. One "definition" of kurtosis is that it is "vaguely ... the location- and scale-free movement of probability mass from the shoulders of a distribution into its center and tails." Here "shoulders" refer to values $\mu \pm \sigma$.

This interpretation suggests that longer tails correspond to more probability within the $\mu$ $\pm$ $\sigma$ range; and conversely, that more probability in the $\mu \pm \sigma$ range implies longer tails. Neither statement is mathematical; simple counterexamples to both of these statements are respectively given as counterexamples 1 and 2 below:

Counterexample 1: $X = \mu + \sigma Z$, where

$Z^2$ = $0.5^2$, with probability (wp) $.50$

= $1.2^2$, wp $0.50 - \theta$

= $0.155/\theta + 1.44$, wp $\theta$.

Take $\pm \sqrt{Z^2}$ with equal probability splits to get the actual $Z$s.

As $\theta \rightarrow 0$, the tail and the kurtosis tend to infinity, but there is always .5 probability within the $\mu \pm \sigma$ range.

Further, as the kurtosis tends to infinity in this family, the "peak" of the distribution becomes more flat-topped, since the probabilities on the four central points all converge to 0.25.

Counterexample 2: $X = \mu + \sigma Z$, where

$Z^2 = \theta$, wp $\theta$

= $2\theta$, wp $(1-\theta)/2$

= $2$, wp $(1-\theta)/2$.

Again, take $\pm \sqrt{Z^2}$ with equal probability splits to get the actual $Z$s.

As $\theta \rightarrow 1$, the probability within the $\mu \pm \sigma$ range tends to 1, but the tail length stays fixed at $\mu + \sqrt{2} \sigma$, and the kurtosis decreases to its minimum, 1.0.

Edit, 9/21/2018: Yet another incorrect myth about kurtosis is that higher kurtosis implies "more probability in the tails." Counterexample 1 above debunks that myth: For that family of distributions, as kurtosis increases, there is less probability in the tails.

Increases in kurtosis imply greater extremity of the tails, not higher probability in the tails. A mathematically precise justification of this statement is given as follows: for any sequence of distributions of random variables $Z$ (wlog having mean 0.0 and variance 1.0) having kurtosis tending to infinity, $E(Z^4 I(|Z| >b))/\text{kurtosis} \rightarrow 1.0$, for every real $b$.


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