Summary

DescriptionPearson type VII distribution PDF.svg	English: 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.
Date	26 May 2020
Source	MarkSweep
Author	Original: MarkSweep   Vector: Andel
Permission (Reusing this file)	Public domainPublic domainfalsefalse This chart is ineligible for copyright and therefore in the public domain, because it consists entirely of information that is common property and contains no original authorship. For more information, see Commons:Threshold of originality § Charts العربية | Deutsch | English | español | français | italiano | 日本語 | македонски | română | русский | slovenščina | 中文（简体） | 中文（繁體） | +/− I, the copyright holder of this work, hereby publish it under the following license:   This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse
Other versions	File:Pearson_type_VII_distribution_log-PDF.svg File:Pearson_type_VII_distribution_PDF.png File:Pearson_type_VII_distribution_log-PDF.png
SVG development InfoField	The SVG code is valid.   This plot was created with Gnuplot.
Source code InfoField	Gnuplot code # the Pearson type VII log-pdf log_p7(x,a2,m) = lgamma(m) - lgamma(m-0.5) - m*log(1+x*x/a2) - 0.5*log(a2*pi) # the Pearson type VII log-pdf with unit variance and kurtosis k f(x,k) = log_p7(x, 2+6.0/k, 2.5+3.0/k) # the standard normal log-pdf (with unit variance and kurtosis 0) n(x) = -0.5 * (x*x + log(2*pi)) # the limit of the Pearson type VII log-pdf for k -> infinity g(x) = -2.5 * log(2 + x*x) + log(3) set samples 1001 set grid set xrange [-10.4:10.4] set xtics 1 set yrange [-0.02:0.57] set ytics (0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.39894, 0.45553, 0.5, 0.53033) set terminal svg size 400,300 enhanced fname 'DejaVu Sans' fsize 10 butt solid set output 'Pearson type VII distribution PDF.svg' plot exp(g(x)) lt 1 lw 2 notitle, \ exp(f(x,2)) lt 3 lw 2 notitle, \ exp(n(x)) lt 7 lw 2 notitle

Background

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

Failed to parse (syntax error): {\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.