ABSTRACT
In many studies e.g. in Basu and Templeman (1984), Zellner and Highfield (1988) and Rockinger and Jondeau (2002) it was shown that the first four moments are sufficient to describe a wide range of distribution types. The formulation using absolute moments in Equation 2 can be modified for the central moments
where the mean value X¯ of a random variable is its first absolute moment. The first central moment µ1 is zero and the second central moment is the variance σ2X . According to Basu and Templeman (1984) the entropy distribution based on central moment constraints reads
where exp (ν0) = 1/c is a constant normalizing the area under the density function. If a standardized random variable is defined
the maximum entropy distribution can be obtained from the standardized central moment constraints
standardized central γ1 and the the kurtosis γ2
From this standardized constraints the distribution parameters can be obtained very efficiently as shown in Rockinger and Jondeau (2002) and van Erp and van Gelder (2008). The final maximum entropy distribution is than obtained for the standardized random variable Y
and finally for the original random variable X as
Special types of the maximum entropy distribution are the uniform distribution (ν′1 = ν′2 = ν′3 = ν′4 = 0, c′ = √12), the exponential distribution (ν′1 = −1, ν′2 = ν′3 = ν′4 = 0, c′ = e) and the normal distribution (ν′2 = −0.5, ν′1 = ν′3 = ν′4 = 0, c′ =
√ 2π).
