Hyperbolic distribution

# Hyperbolic distribution

Discussion

Encyclopedia
The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

is a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

. Thus the distribution decreases exponentially, which is more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The hyperbolic distributions form a subclass of the generalised hyperbolic distributions.

The origin of the distribution is the observation by Ralph Alger Bagnold
Ralph Alger Bagnold
Brigadier Ralph Alger Bagnold, FRS OBE, was the founder and first commander of the British Army's Long Range Desert Group during World War II. He is also generally considered to have been a pioneer of desert exploration, an acclaim earned for his activities during the 1930s...

in his book The Physics of Blown Sand and Desert Dunes (1941) that the logarithm of the histogram of the empirical size distribution of sand deposits tends to form a hyperbola. This observation was formalised mathematically by Ole Barndorff-Nielsen in a paper in 1977, where he also introduced the generalised hyperbolic distribution, using the fact the a hyperbolic distribution is a random mixture of normal distributions.