Kurtosis - Kurtosis of Well-known Distributions

Kurtosis of Well-known Distributions

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, 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, whereas the densities with positive or zero excess kurtosis are supported on the whole real line.

There exist platykurtic densities with infinite support,

• e.g., exponential power distributions 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