quality:correction, stylistic:tone
The Pearson type VII family

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 type VII family, which is a special case of the Pearson type IV family restricted to symmetric densities. The probability density function is given by f (x; a, m) = Γ (m) a π Γ (m − 1 / 2) − m, {\displaystyle f(x;a,m)={\frac {\Gamma (m)}{a\,{\sqrt {\pi }}\,\Gamma (m-1/2)}}\left^{-m},} where a is a scale parameter and m is a shape parameter.

All densities in this family are symmetric. The k th moment exists provided m > (k + 1)/2. For the kurtosis to exist, we require m > 5/2. Then the mean and skewness exist and are both identically zero. Setting a = 2 m − 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 m = 5 / 2 + 3 / γ 2 {\displaystyle m=5/2+3/\gamma _{2}}, where γ 2 {\displaystyle \gamma _{2}} 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 g (x; γ 2) = f (x; a = 2 + 6 γ 2, m = 5 2 + 3 γ 2). {\displaystyle g(x;\gamma _{2})=f\left(x;\;a={\sqrt {2+{\frac {6}{\gamma _{2}}}}},\;m={\frac {5}{2}}+{\frac {3}{\gamma _{2}}}\right).}

In the limit as γ 2 → ∞ {\displaystyle \gamma _{2}\to \infty } one obtains the density g (x) = 3 (2 + x 2) − 5 2, {\displaystyle g(x)=3\left(2+x^{2}\right)^{-{\frac {5}{2}}},} which is shown as the red curve in the images on the right.

In the other direction as γ 2 → 0 {\displaystyle \gamma _{2}\to 0} one obtains the standard normal density as the limiting distribution, shown as the black curve.

In the images on the right, the blue curve represents the density x ↦ g (x; 2) {\displaystyle x\mapsto g(x;2)} 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 γ = 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 γ 2 = ∞ {\displaystyle \gamma _{2}=\infty } (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").