Generalized normal distribution and kurtosis

The generalized normal distribution adds an extra parameter β to the normal (Gaussian) distribution. The probability density function for the generalized normal distribution is

\frac{\beta}{2\sigma \Gamma\left(\frac{1}{\beta}\right )} \exp\left(-\left|\frac{x-\mu}{\sigma} \right|^\beta \right)

Here the location parameter μ is the mean, but the scaling factor σ is not the standard deviation unless β = 2.

For small values of the shape parameter β, the distribution is more sharply pointed in the middle and decays more slowly in the tails. We say the tails are “thick” or “heavy.” When β = 1 the generalized normal distribution reduces to the Laplace distribution.

Here are examples with μ = 0 and σ = 1.

The normal distribution is a special case corresponding to β = 2. Large values of β make the distribution flatter on top and thinner (lighter) in the tails. Again μ = 0 and σ = 1 in the plots below.

thick-tailed generalized normal densities

One way to measure the thickness of probability distribution tails is kurtosis. The normal distribution has kurtosis equal to 3. Smaller values of kurtosis correspond to thinner tails and larger values to thicker tails.

There’s a common misunderstanding that kurtosis measures how pointy the distribution is in the middle. Often that’s the case, and in fact that’s the case for the generalized normal distribution. But it’s not true in general. It’s possible for a distribution to be flat on top and have heavy tails or pointy on top and have thin tails.

Distributions with thinner tails than the normal are called “platykurtic” and distributions with thicker tails than the normal are called “leptokurtic.” The names were based on the misunderstanding mentioned above. The platy– prefix means broad, but it’s not the tails that are broader, it’s the middle. Similarly, the lepto– prefix means “thin”, referring to being pointy in the middle. But leptokurtic distributions have thicker tails!

The kurtosis of the generalized normal distribution is given by

\frac{ \Gamma\left( \frac{5}{\beta} \right ) \Gamma\left( \frac{1}{\beta} \right ) }{\Gamma\left(\frac{3}{\beta}\right)^2}

We can use that to visualize how the kurtosis varies as a function of the shape parameter β.

The Laplace distribution (β = 1) has kurtosis 6 and the normal distribution (β = 2) has kurtosis 3.

You can use the fact that Γ(x) ~ 1/x for small x to show that in the limit as β goes to infinity, the kurtosis approaches 9/5.

Related post: Computing skewness and kurtosis in one pass

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