Probability density function of the Pearson type VII distribution

The red curve shows the limiting density with infinite kurtosis; the blue curve shows the density with kurtosis equal to 2; the black curve shows the limiting (normal) density with kurtosis identically zero.

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The Pearson type VII family of probability densities is a special case of the type IV family restricted to symmetric densities. The probability density function is given by

${\displaystyle f(x;a^{2},m)={\frac {\Gamma (m)}{\Gamma (m-1/2)\,{\sqrt {a^{2}\,\pi }}}}\left(1+{\frac {x^{2}}{a^{2}}}\right)^{-m}.\!}$

All densities in this family are symmetric with zero mean. Setting ${\displaystyle a^{2}=2m-3}$ makes the variance equal to unity. Then the only free parameter is m, which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize with ${\displaystyle m=5/2+3/k}$ where k is the kurtosis to obtain a one-parameter leptokurtic family with zero mean, unit variance, zero skew, and arbitrary positive kurtosis k.

In the limit as ${\displaystyle k\to \infty }$ one obtains the density

${\displaystyle g(x)=3\left(2+x^{2}\right)^{-5/2}\!}$

shown as the red curve. In the other direction as ${\displaystyle k\to 0}$ one obtains the standard normal density as the limiting distribution, shown as the black curve.

 

This plot was created with Gnuplot by n.

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