# MeixnerDistribution

MeixnerDistribution[a,b,m,d]

represents a Meixner distribution with location parameter m, scale parameter a, skew parameter b, and shape parameter d.

# Details • The probability density for value in a Meixner distribution is proportional to .
• MeixnerDistribution allows m to be any real number, a and d to be any positive real number, and b such that .
• MeixnerDistribution allows m and a to be any quantities of the same unit dimensions, and b and d to be dimensionless quantities. »
• MeixnerDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• MeixnerDistribution[a,b,m,d] represents a continuous statistical distribution defined and supported over the set of real numbers and parametrized by a real number m (called a "location parameter"), two positive real numbers a and d (a "scale parameter" and a "shape parameter", respectively), and a real number (called a "skew parameter"). Together, these parameters determine the overall behavior of its probability density function (PDF). In general, the PDF of a Meixner distribution is unimodal with a single "peak" (i.e. a global maximum) though its overall shape (height, spread, and horizontal location of its maximum) is determined by the values of a, b, d, and m. In addition, the PDF of the Meixner distribution has tails that are "semiheavy", which roughly means that for large values of , its PDF decreases somewhat more quickly than algebraically but considerably less so than exponentially. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
• The origins of the Meixner distribution lie in German physicist Josef Meixner's 1930s work on generating functions for systems of orthogonal polynomials in statistical mechanics. The Meixner distribution belongs to the family of so-called generalized -distributions, which is a set of probability distributions satisfying certain properties desirable in the construction and study of Lévy processes. In particular, the Meixner distribution is the foundation of the so-called Meixner process, a stochastic process starting at 0 and featuring independent and stationary increments that are all Meixner-distributed. The Meixner process (and hence the Meixner distribution) is a widely used tool in finance, particularly in modeling derivative pricing, though the distribution itself also arises in the study of various topics, including random matrices.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Meixner distribution. Distributed[x,MeixnerDistribution[a,b,m,d]], written more concisely as xMeixnerDistribution[a,b,m,d], can be used to assert that a random variable x is distributed according to a Meixner distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for Meixner distributions may be given using PDF[MeixnerDistribution[a,b,m,d],x] and CDF[MeixnerDistribution[a,b,m,d],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a Meixner distribution, EstimatedDistribution to estimate a Meixner parametric distribution from given data, and FindDistributionParameters to fit data to a Meixner distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Meixner distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Meixner distribution.
• TransformedDistribution can be used to represent a transformed Meixner distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Meixner distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Meixner distributions.
• MeixnerDistribution is related to a number of other distributions. Qualitatively, generalized -distributions such as MeixnerDistribution and FisherZDistribution are defined to share analytic properties of the gamma function. Similarly, MeixnerDistribution is qualitatively related to InverseGaussianDistribution, HyperbolicDistribution, GammaDistribution, and VarianceGammaDistribution. Quantitatively, MeixnerDistribution generalizes SechDistribution in the sense that the PDF of SechDistribution[μ,σ/2] is precisely the same as that of MeixnerDistribution[σ,0,μ,1/2].

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median is available numerically:

## Scope(8)

Generate a sample of pseudorandom numbers from a Meixner distribution:

Compare the histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness and kurtosis:

The distribution becomes more symmetric when d is large:

Kurtosis:

In the limit, the distribution is mesokurtic:

Different moments with closed forms as functions of parameters:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the mean thermal expansion coefficient:

## Applications(1)

MeixnerDistribution can be used to model the returns of the S&P 500:

Fit Meixner distribution to the data:

Compare the histogram of the data with the PDF of the estimated distribution:

## Properties & Relations(4)

Relationships to other distributions:

SechDistribution is a special case of MeixnerDistribution:

The Ito integral follows a Meixner distribution:

Check that the low-order moments are the same:

Sample from process values at :

Check the goodness of fit with MeixnerDistribution:

Confirm the fit visually:

## Neat Examples(1)

PDFs for different b values with CDF contours: