represents the stable distribution Stype with index of stability α, skewness parameter β, location parameter μ, and scale parameter σ.
- A linear combination of independent identically distributed stable random variables is also stable.
- A stable distribution is defined in terms of its characteristic function , which satisfies a functional equation where for any and there exist and such that . The general solution to the functional equation has four parameters.
- StableDistribution allows 0<α≤2, -1≤β≤1, μ to be any real number, and σ to be any positive real number.
- StableDistribution allows μ and σ to be any quantities of the same unit dimensions, and α, β to be dimensionless quantities. »
- CharacteristicFunction[StableDistribution[0,α,…],t] is continuous in α and given by .
- CharacteristicFunction[StableDistribution[1,α,…],t] is discontinuous in α and given by .
- StableDistribution[α] is equivalent to StableDistribution[1,α,0,0,1].
- StableDistribution[α,β] is equivalent to StableDistribution[1,α,β,0,1].
- StableDistribution[α,β,μ,σ] is equivalent to StableDistribution[1,α,β,μ,σ].
- StableDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- StableDistribution[type,α,β,μ,σ] represents a continuous statistical distribution belonging to one of two types and parametrized by the positive real number σ (called a "scale parameter") and by the real numbers μ (a "location parameter"), α (the index of stability of the distribution for which ), and β (a "skewness parameter" satisfying ), which together determine the overall behavior of its probability density function (PDF).
- In general, the PDF of a stable distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined both by its type and by the values of α, β, μ, and σ. In addition, the tails of the PDF may be "fat" (i.e. the PDF decreases non-exponentially for large values ) or "thin" (i.e. the PDF decreases exponentially for large ), depending on the values of type, α, β, μ, and σ. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The four-, two-, and one-parameter versions StableDistribution[α,β,μ,σ], StableDistribution[α,β], and StableDistribution[α] are equivalent to the type-1 distributions StableDistribution[1,α,β,μ,σ], StableDistribution[1,α,β,0,1], and StableDistribution[1,α,0,0,1], respectively. For various values of its parameters, the stable distribution may be referred to as the stable-Paretian distribution, the Pareto–Lévy distribution (not to be confused with ParetoDistribution or LevyDistribution), or as the Lévy α-stable distribution. The stable distribution is also distinct from the similarly named min- and max-stable distributions (MinStableDistribution and MaxStableDistribution, respectively).
- Despite many special cases of the stable distribution being classical in nature, the family of stable distributions described above was first studied by mathematician Paul Lévy in the mid-1920s. The family of stable distributions is characterized by being closed under linear combinations, meaning their PDFs in general do not have closed-form expression and must instead be described in terms of their characteristic functions (CharacteristicFunction). Stable distributions are particularly important in stochastics and probability due to the role they play in generalizing the so-called central limit theorem, though such distributions have also been used to model phenomena in finance, astronomy, and physics.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a stable distribution. Distributed[x,StableDistribution[type,α,β,μ,σ]], written more concisely as xStableDistribution[type,α,β,μ,σ], can be used to assert that a random variable x is distributed according to a stable distribution of a given type. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for stable distributions of a given type may be given using PDF[StableDistribution[type,α,β,μ,σ],x] and CDF[StableDistribution[type,α,β,μ,σ],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 stable distribution, EstimatedDistribution to estimate a stable parametric distribution from given data, and FindDistributionParameters to fit data to a stable distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic stable distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic stable distribution.
- TransformedDistribution can be used to represent a transformed stable 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 stable distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving stable distributions.
- StableDistribution is closely related to a number of other distributions. LandauDistribution, CauchyDistribution, NormalDistribution, and LevyDistribution are examples of type-1 stable distributions in the sense that LandauDistribution[μ,σ] has the same characteristic function (CharacteristicFunction) as StableDistribution[1,1,1,μ,σ], CauchyDistribution[0,1] has the same PDF as StableDistribution[1,1,0,0,1], the PDF of NormalDistribution[μ,σ] is equivalent to that of StableDistribution[1,2,β,μ,σ/], and the PDF of LevyDistribution[μ,σ] is precisely the same as that of StableDistribution[1,1/2,1,μ,σ]. Qualitatively, StableDistribution is similar to PearsonDistribution, and it is also closely related to ParetoDistribution, BetaDistribution, GammaDistribution, and HalfNormalDistribution.
Examplesopen allclose all
Basic Examples (4)
Map–Airy distribution [MathWorld] is a member of the stable family:
Generalized central limit theorem gives sequences and such that the distribution of the shifted and rescaled sum of i.i.d. random variates whose distribution function has asymptotes as and as weakly converges to the stable distribution :
Properties & Relations (11)
LandauDistribution is a stable distribution:
CauchyDistribution is a stable distribution:
NormalDistribution is a stable distribution:
LevyDistribution is a stable distribution:
Wolfram Research (2010), StableDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StableDistribution.html (updated 2016).
Wolfram Language. 2010. "StableDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/StableDistribution.html.
Wolfram Language. (2010). StableDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StableDistribution.html