Editing Kurtosis (section)

== Graphical examples ==

=== The Pearson type VII family ===
[[Image:Pearson type VII distribution PDF.svg|300px|thumb|[[Probability density function|pdf]] for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); and 0 (black)]]
[[Image:Pearson type VII distribution log-PDF.svg|300px|thumb|log-pdf for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)]]

The effects of kurtosis are illustrated using a [[parametric family]] of distributions whose kurtosis can be adjusted while their lower-order moments and cumulants remain constant. Consider the [[Pearson distribution|Pearson type VII family]], which is a special case of the [[Pearson distribution|Pearson type IV family]] restricted to symmetric densities. The [[probability density function]] is given by
<math display="block">f(x; a, m) = \frac{\Gamma(m)}{a\,\sqrt{\pi}\,\Gamma(m-1/2)} \left[1+\left(\frac{x}{a}\right)^2 \right]^{-m}, </math>
where {{mvar|a}} is a [[scale parameter]] and {{mvar|m}} is a [[shape parameter]].

All densities in this family are symmetric. The {{mvar|k}}-th moment exists provided {{math|''m'' > (''k'' + 1)/2}}. For the kurtosis to exist, we require {{math|''m'' > 5/2}}. Then the mean and [[skewness]] exist and are both identically zero. Setting {{math|1=''a''<sup>2</sup> = 2''m'' − 3}} makes the variance equal to unity. Then the only free parameter is {{mvar|m}}, which controls the fourth moment (and cumulant) and hence the kurtosis.  One can reparameterize with <math display="inline">m = 5/2 + 3/\gamma_2</math>, where <math>\gamma_2</math> is the excess kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary non-negative excess kurtosis. The reparameterized density is
<math display="block">g(x; \gamma_2) = f{\left(x;\; a = \sqrt{2 + 6 \gamma_2^{-1}},\; m = \tfrac{5}{2} + 3\gamma_2^{-1} \right)}. </math>

In the limit as <math>\gamma_2 \to \infty</math> one obtains the density
<math display="block">g(x) = 3\left(2 + x^2\right)^{-5/2}, </math>
which is shown as the red curve in the images on the right.

In the other direction as <math>\gamma_2 \to 0</math> one obtains the [[normal distribution|standard normal]] density as the limiting distribution, shown as the black curve.

In the images on the right, the blue curve represents the density <math>x \mapsto g(x; 2)</math> with excess kurtosis of 2.  The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density, although this conclusion is only valid for this select family of distributions. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is a [[parabola]]. One can see that the normal density allocates little probability mass to the regions far from the mean ("has thin tails"), compared with the blue curve of the leptokurtic Pearson type VII density with excess kurtosis of 2.  Between the blue curve and the black are other Pearson type VII densities with {{math|''γ''<sub>2</sub>}}&nbsp;=&nbsp;1, 1/2, 1/4, 1/8, and 1/16.  The red curve again shows the upper limit of the Pearson type VII family, with <math>\gamma_2 = \infty</math> (which, strictly speaking, means that the fourth moment does not exist). The red curve decreases the slowest as one moves outward from the origin ("has fat tails").
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=== Other well-known distributions ===
[[Image:Standard symmetric pdfs.svg|300px|thumb|[[Probability density function]]s for selected distributions with [[expected value|mean]] 0, [[variance]] 1 and different excess kurtosis]]
[[Image:Standard symmetric pdfs logscale.svg|300px|thumb|[[Logarithm]]s of [[probability density function]]s for selected distributions with [[expected value|mean]] 0, [[variance]] 1 and different excess kurtosis]]

Several well-known, unimodal, and symmetric distributions from different parametric families are compared here.  Each has a mean and skewness of zero. The parameters have been chosen to result in a variance equal to 1 in each case. The images on the right show curves for the following seven densities, on a [[linear scale]] and [[logarithmic scale]]:

* D: [[Laplace distribution]], also known as the double exponential distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3
* S: [[hyperbolic secant distribution]], orange curve, excess kurtosis = 2
* L: [[logistic distribution]], green curve, excess kurtosis = 1.2
* N: [[normal distribution]], black curve (inverted parabola in the log-scale plot), excess kurtosis = 0
* C: [[raised cosine distribution]], cyan curve, excess kurtosis = −0.593762...
* W: [[Wigner semicircle distribution]], blue curve, excess kurtosis = −1
* U: [[uniform distribution (continuous)|uniform distribution]], magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.

Note that in these cases the platykurtic densities have bounded [[Support (mathematics)|support]], whereas the densities with positive or zero excess kurtosis are supported on the whole [[real line]].

One cannot infer that high or low kurtosis distributions have the characteristics indicated by these examples. There exist platykurtic densities with infinite support,
*e.g., [[exponential power distribution]]s with sufficiently large shape parameter ''b''

and there exist leptokurtic densities with finite support.
*e.g., a distribution that is uniform between −3 and −0.3, between −0.3 and 0.3, and between 0.3 and 3, with the same density in the (−3, −0.3) and (0.3, 3) intervals, but with 20 times more density in the (−0.3, 0.3) interval

Also, there exist platykurtic densities with infinite peakedness,
*e.g., an equal mixture of the [[beta distribution]] with parameters 0.5 and 1 with its reflection about 0.0

and there exist leptokurtic densities that appear flat-topped,
*e.g., a mixture of distribution that is uniform between −1 and 1 with a T(4.0000001) [[Student's t-distribution]], with mixing probabilities 0.999 and 0.001.

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