"Peakedness" of a skewed probability density function

Question

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?

Answer 1

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).

Answer 2

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.

Answer 3

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, $w_2=\frac{\tilde{\Phi_{99}}-\tilde{\Phi_{50}}}{\tilde{\Phi_{75}}-\tilde{\Phi_{50}}}$, $\tau=\frac{w_1}{w_2}$.

Answer 4

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

See BigBendRegion's answer. Also read his relevant paper.

The features I want to describe, would they be called "kurtosis"?

What's in a word? The excess kurtosis is given by

$$g_2 = \frac{\frac{1}{m} \sum_{i=1}^m (x- \bar x)^4}{\left[ \frac{1}{m} \sum_{i=1}^m (x_i - \bar x)^2 \right]^2} - 3$$

and has the upper bound

$$g_2 \leq \frac{1}{2} \frac{m-3}{m-2} g_1^2 + \frac{m}{2} - 3 = B$$

where $g_1$ is the similarly-defined skewness. Finding a dataset which minimized $\frac{g_2}{B}$ I found:

And finding a similarly maximal example:

These patterns are unusual, and not what I would intuitively agree to call minimal or maximal "peakedness".

I've only seen the word "kurtosis" used for symmetric distributions?

(Pearson's) kurtosis requires that the fourth moments exist, which is true of many distributions. Some of them even have a really clean expression, such as the excess kurtosis for a gamma distribution.

Answer 5

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?