Error distribution


Format: Error(m, s, n)

 

The Error distribution goes by variety of names:

Exponential Power Distribution

Generalized Error Distribution (GED)

Generalized Gaussian distribution (GGD)

Subbotin distribution

To add to the confusion, you will also see a wide range of parameterizations. In ModelRisk, we have chosen to use the mean m, standard deviation s and power index n to parameterize the distribution, because it make comparisons with the Normal, Laplace, Hyperbolic-Secant and other symmetric distributions easier.

This three parameter distribution offers a variety of symmetric shapes, as shown in the figures below. The first pane shows the effect on the distribution's shape of varying parameter n . Note n  = 2 is a Normal distribution, n =1 is a Laplace distribution and the distribution approaches a Uniform as n approaches infinity. The second pane shows the change in the distribution's spread by varying parameter s, its standard deviation. Parameter m is simply the location of the distribution's peak, and the distribution's mean.

 

Uses

The Error distribution finds quite a lot of use as a prior distribution in Bayesian inference because it has greater flexibility than a Normal prior, in that the Error distribution is flatter than a Normal (platykurtic) when n > 2, and more peaked than a Normal distribution (leptokurtic) when n < 2. Thus, using the GED allows one to maintain the same mean and variance, but vary the distribution's shape (via the parameter n) as required.

The Error distribution has also used to model variations in historic UK property market returns.

The 'Error Function' distribution, distinct from the distribution described here, is another format for the Normal distribution with a zero mean, i.e. Erf(h) = Normal(0, 1/(h*SQRT(2)))

ModelRisk functions added to Microsoft Excel for the Error distribution

VoseError generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseErrorObject constructs a distribution object for this distribution.

VoseErrorProb returns the probability density or cumulative distribution function for this distribution.

VoseErrorProb10 returns the log10 of the probability density or cumulative distribution function.  

VoseErrorFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseErrorFitObject constructs a distribution object of this distribution fitted to data.

VoseErrorFitP returns the parameters of this distribution fitted to data.

 

Error distribution equations

 

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