# “Peakedness” of a skewed probability density function

I would like to describe the "peakedness" and tail "heaviness" of several skewed probability density functions.

The features I want to describe, would they be called "kurtosis"? I've only seen the word "kurtosis" used for symmetric distributions?

• Indeed, the measures of kurtosis are typically applied to symmetric distributions. You can calculate it for skewed ones as well but the interpretation changes since this value varies when the asymmetry is introduced. In fact, these two concepts are difficult to separate. Recently, a skewness-invariant measure of kurtosis was proposed in this paper. – user10525 Jan 3 '13 at 16:22
• High kurtosis is associated with peakedness and with heavy tailedness (it's also characterized as 'lack of shoulders'). One of the volumes of Kendall and Stuart discuss these issues at some length. But such interpretations, are, as you note, generally given in the situation of near-symmetry. In nonsymmetric cases, the standardized 4th moment is usually highly correlated with the square of the standardized third moment, so they're mostly measuring much the same kind of thing. – Glen_b -Reinstate Monica Jan 3 '13 at 23:44
• Indeed, given the particular way I phrased it in my earlier comment, it's true even of symmetric distributions - the square of the sample standardized third moment (squared moment skewness) is highly correlated with the sample standardized fourth moment ('kurtosis'), even at say the normal. – Glen_b -Reinstate Monica Jun 19 '13 at 0:27

With variance being defined as the second moment $\mu_{2}$, skewness being defined as the third moment $\mu_{3}$ and the kurtosis being defined as the fourth moment $\mu_{4}$, it is possible to describe the properties of a wide range of symmetric and non-symmetric distributions from the data.

This technique was originally described by Karl Pearson in 1895 for the so-called Pearson Distributions I to VII. This has been extended by Egon S Pearson (date uncertain) as published in Hahn and Shapiro in 1966 to a wide range of symmetric, asymmetric and heavy tailed distributions that include Uniform, Normal, Students-t, Lognormal, Exponential, Gamma, Beta, Beta J and Beta U. From the chart of p. 197 of Hahn and Shapiro, $B_{1}$ and $B_{2}$ can be used to establish descriptors for skewness and kurtosis as:

$\mu_{3} = \sqrt {B_{1}\ \mu_{2}^{3}}$
$\mu_{4} = B_{2}\ \mu_{2}^{2}$

If you just wanted simple relative descriptors then by applying a constant $\mu_{2} = 1$ the skewness is $\sqrt {B_{1}}$ and the kurtosis is $B_{2}$.

We have attempted to summarize this chart here so that it could be programmed, but it is better to review it in Hahn and Shapiro (pp 42-49,122-132,197). In a sense we are suggesting a little bit of reverse engineering of the Pearson chart, but this could be a way to quantify what you are seeking.

The main issue here is, what is "peakedness"? Is it curvature at the peak (2nd derivative?) Does it require standardization first? (You would think so, but there is a stream of literature starting with Proschan, Ann. Math. Statist. Volume 36, Number 6 (1965), 1703-1706, that defines peakedness in a way that normal with smaller variance are more "peaked"). Or is it probability concentration within a standard deviation of the mean, as implicit in Balanda and Macgillivray (The American Statistician, 1988, Vol 42, 111-119)? Once you settle on a definition, then it should be trivial to apply it. But I would ask, "why do you care?" Of what relevance is "peakedness", however defined?

BTW, Pearson's kurtosis measures tails only, and does not measure any of the above mentioned "peakedness" definitions. You can change the data or distribution within a standard deviation of mean as much as you want (keeping the mean=0 and variance=1 constraint), but the kurtosis can only change within a maximum range of 0.25 (usually much less). So you can rule out using kurtosis to measure peakedness for any distribution, even though kurtosis is indeed a measure of tails for any distribution, no matter whether the distribution is symmetric, asymmetric, discrete, continuous, discrete/continuous mixture, or empirical. Kurtosis measures tails for all distributions, and virtually nothing about peak (however defined).

A possible very practical approach could be calculate the ratio of the survival function of the distribution $\Pr\left(\tilde X \gt 1- \alpha \right)$ against the normal one, showing it is quite far greater. Another approach can be calculating the ratios of percentiles $w_1=\frac{\tilde{x_{99}}-\tilde{x_{50}}}{\tilde{x_{75}}-\tilde{x_{50}}}$ of the distribution $\tilde x$ under interest and dividing it against the normal one quantile values, $w_2=\frac{\tilde{\Phi_{99}}-\tilde{\Phi_{50}}}{\tilde{\Phi_{75}}-\tilde{\Phi_{50}}}$, $\tau=\frac{w_1}{w_2}$.

I am not sure I get your understanding of peakedness and heaviness. Kurtosis means "Excess" in German, so it describes the "head" or "peak" of a distribution, describing whether it is very wide or very narrow. Wikipedia states that the "peakedness" is actually described by the "kurtosis", whereas peakedness does not to appear to be a real word and you should use the term "Kurtosis".

So I think you might have gotten everything right, the head is the Kurtosis, The "heaviness" of the tail might be the Skewness":

Here is how you find it:

$$a_3 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^3}{N * s^3_x}$$

with s as the standard deviation for x.

The values indicate:

Negative Skew: $$a_3 < 0$$

Positive Skew: $$a_3 > 0$$

No Skew $$a_3 = 0$$

You can get a value for the kurtosis with: $$a_4 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^4}{N * s^4_x}$$

The values indicate:

Platycurtic: $$a_4 < 3$$

Leptocurtic: $$a_4 > 3$$

Normal: $$a_4 = 3.0$$

Did that help?

• I'm afraid this answer in its current form may be less than helpful due to errors in it. Skewness is a standard measure of asymmetry. It is not closely related to heaviness of tails: it is possible for the tails to be extremely heavy and the skewness to be zero (which is the case for any symmetric distribution, for instance). Please note, too, that it is impossible for $a_4$ to be negative, so the second half of this answer makes little sense. (Perhaps you confused kurtosis with excess kurtosis?) – whuber Jan 3 '13 at 17:53
• Thank you for clarifying. There might indeed be some errors in the formulas, I just copied them from the scripts they provide at uni. I oversaw the fact that a4 can't be negative. – Johannes Hofmeister Jan 3 '13 at 22:46
• I looked up why my answer is wrong - it is a translational error, I apologize for that. My slides are all in German, mixing Kurtosis and Excess. – Johannes Hofmeister Jan 3 '13 at 22:52
• @Peter As Peter Westfall keeps pointing out, your comment is incorrect: "peakedness" (of any mode), thought of vaguely as pointiness or height, has absolutely nothing to do with the tails of any distribution, nor is it measured by any finite combination of moments (such as the kurtosis). It may happen to be connected to heaviness of tails for a family of distributions, but that's a completely different matter. – whuber Jan 3 '18 at 13:44

Kurtosis is definitely associated with the peakedness of the curve. I henceforth believe that you are really looking for kurtosis which does exist whether the distribution is symmetric or not. (user10525) has definitely said it right ! I hope your problem is resolved by now. Do share its outcome, all opinions are welcome.

• I'm not sure how this constitutes a helpful answer beyond what was already written here. How about you expand more on kurtosis and peakedness of the curve? – Momo Oct 18 '13 at 18:58
• Wanted to give clear cut clarification to the query. The discussion seemed to be confusing @Momo – Vani Oct 18 '13 at 19:00