StableDistribution

✖
`StableDistribution`

✖

StableDistribution[type,α,β,μ,σ]

represents the stable distribution S_type with index of stability α, skewness parameter β, location parameter μ, and scale parameter σ.

Details

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 $exp(i mu t-sigma TemplateBox[{t}, Abs] (1+2i beta/pi sgn(t) log(TemplateBox[{{t, , sigma}}, Abs]))) alpha=1; exp(i mu t-TemplateBox[{{t, , sigma}}, Abs]^alpha (1+i beta tan((pi alpha)/2) sgn(t) (TemplateBox[{{t, , sigma}}, Abs]^(1-alpha)-1))) alpha!=1$ .
CharacteristicFunction[StableDistribution[1,α,…],t] is discontinuous in α and given by $exp(i mu t-sigma TemplateBox[{t}, Abs] (1+2 i beta/pi sgn(t) log(TemplateBox[{t}, Abs]))) alpha=1; exp(i mu t-TemplateBox[{{t, , sigma}}, Abs]^alpha (1-i beta tan((pi alpha)/2) sgn(t))) alpha!=1$ .
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 xStableDistribution[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.

Examples

open allclose all

Basic Examples (4)Summary of the most common use cases

Probability density function for type 1 for a range of skewness parameters:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-61u8pd

Out[1]=1

Probability density function for type 0 for various stability indexes:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-22y84s

Out[2]=2

Cumulative distribution function for type 1:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-0k63lk

Out[1]=1

Type 0:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-yc7jct

Out[2]=2

Mean depends on the type:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-1rsxul

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-9imddx

Out[2]=2

Variance is type independent and is only defined for :

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-2hgob1

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-y4ayui

Out[2]=2

Scope (4)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-tscvhm

Compare its histogram to the PDF:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-fw0ala

Out[2]=2

Higher moments are only defined for :

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-lpkqey

Out[1]=1

This is the case where StableDistribution reduces to NormalDistribution:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-c1ic0u

Out[2]=2

Hazard function for different stability indexes:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-rwxqe

Out[1]=1

Hazard function for different skewness parameters:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-i4zm47

Out[2]=2

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-bsnti9

Out[1]=1

Find the 75% quantile:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-gpku4f

Out[2]=2

Applications (9)Sample problems that can be solved with this function

Assuming daily logarithmic return of the stock market follows a stable distribution, simulate and visualize stock prices over a period of 5 years:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-lkl8l5

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-wb23d

Out[2]=2

Assuming stock logarithmic return follows a stable distribution, find the value at risk at the 95% level:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-k7g4dd

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-czspj

Out[2]=2

Compute the 95% value at risk point loss of the current S&P 500 index value, assuming the above distribution:

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-4lmkm

Out[3]=3

Find the expected shortfall of logarithmic return:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-c0tv9a

Out[4]=4

Compute the associated point loss:

In[5]:=5

✖

https://wolfram.com/xid/0bnhgp071je6i-c6jv19

Out[5]=5

Fit the daily logarithm return of the IBM stocks since January 1, 2005 to a stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-s09vhx

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-b4vvm

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-edue1i

Out[3]=3

Compute logarithmic returns:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-ktsk4f

Fit logarithmic returns to a stable distribution:

In[5]:=5

✖

https://wolfram.com/xid/0bnhgp071je6i-jzpyl8

Out[5]=5

Compare the estimated distribution to a data histogram:

In[6]:=6

✖

https://wolfram.com/xid/0bnhgp071je6i-fhjcvg

Out[6]=6

The product of a symmetric stable random variate and the power of an exponential random variate follows a Linnik distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-ijc04v

Calculate the characteristic function of a Linnik distribution:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-cwesfy

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-gdbs2u

Out[3]=3

Generate random variates and show the histogram:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-fqw4d7

Out[4]=4

Map–Airy distribution [MathWorld] is a member of the stable family:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-gmjmf1

Its probability density is known in closed form:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-d9dqr0

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-jq38zm

Out[3]=3

Find the location of the mode:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-db1d7g

Out[4]=4

Estimate the parameter of stable distribution from a sample characteristic function:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-mz1ysf

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-pug6py

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-d1bcov

Out[3]=3

Plot absolute values of a sample characteristic function and a population characteristic function:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-0l8i7

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0bnhgp071je6i-by8u9u

Out[5]=5

Compare with the maximum likelihood estimation:

In[6]:=6

✖

https://wolfram.com/xid/0bnhgp071je6i-n2cz0t

Out[6]=6

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 :

Illustrate the generalized central limit theorem using a two-sided Pareto distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-sq9b4

Define the mean and variance of the two-sided Pareto distribution for future use:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-bsm74x

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-bn85jc

Define a routine to generate -variates:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-gezeu4

Define a function to visualize a density plot and data histogram:

In[5]:=5

✖

https://wolfram.com/xid/0bnhgp071je6i-elofwm

Case of :

In[6]:=6

✖

https://wolfram.com/xid/0bnhgp071je6i-phvlws

In[7]:=7

✖

https://wolfram.com/xid/0bnhgp071je6i-qdzygf

Out[7]=7

Case of :

In[8]:=8

✖

https://wolfram.com/xid/0bnhgp071je6i-d14fbw

In[9]:=9

✖

https://wolfram.com/xid/0bnhgp071je6i-nnv9t8

Out[9]=9

Case of :

In[10]:=10

✖

https://wolfram.com/xid/0bnhgp071je6i-exvt0g

In[11]:=11

✖

https://wolfram.com/xid/0bnhgp071je6i-jpzlvt

Out[11]=11

Case of :

In[12]:=12

✖

https://wolfram.com/xid/0bnhgp071je6i-b6w3ma

In[13]:=13

✖

https://wolfram.com/xid/0bnhgp071je6i-xacmu

Out[13]=13

Case of standard central limit theorem, i.e. :

In[14]:=14

✖

https://wolfram.com/xid/0bnhgp071je6i-hph37q

In[15]:=15

✖

https://wolfram.com/xid/0bnhgp071je6i-e1cdcu

Out[15]=15

Holtsmark distribution is the distribution of forces acting on a particle in an infinite Poisson system. The ‐component of the gravitational force follows symmetric stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-xs481

Simulate absolute value of the force:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-d4rcpp

Out[2]=2

Distribution of the absolute value is known in closed form:

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-g326sw

Out[3]=3

Compare it to the histogram:

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-f750vn

Out[4]=4

A maximally right‐skewed stable variate, i.e. , is often referred to as positive stable:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-h2j68t

The Laplace transform of a positive stable density has the simple form :

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-izny6k

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-m5uzpa

Out[3]=3

Properties & Relations (11)Properties of the function, and connections to other functions

Stable variables of type 0 and type 1 are related to each other by a shift of location parameter:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-ciyl55

Verify using characteristic function:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-f5rro3

Out[2]=2

And for type 1:

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-bcaisu

Out[3]=3

Discontinuity in of an stable random variate is manifested in the sensitivity of mode to small changes in :

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-bzsp8t

Out[4]=4

A family of stable distributions of type 0 is closed under shifting and scaling:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-zggnk

Out[1]=1

The proof uses characteristic functions:

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-2zpwj8

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-l3d6ze

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-bcsmhy

Out[4]=4

A family of stable distributions of type 1 is closed under shifting and scaling:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-nugwvo

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-zxuzlv

Out[2]=2

Sum of two stable variates of the same stability index is again a stable variate:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-j6wn66

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-koa49i

Considering first the case when is not 1:

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-cygyk3

Out[3]=3

And for :

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-fe2r2x

Out[4]=4

Strictly stable distributions satisfy duality law, for and :

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-btsal4

Dual strictly stable distribution with stability index :

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-bjf76w

The duality law states that for , the following equality holds:

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-lous53

Out[3]=3

Stable variates with are stochastically ordered, i.e. implies :

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-dxar4h

Out[1]=1

Relationships to other distributions:

LandauDistribution is a stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-6slhh0

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-i86myc

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-7q2hyd

Out[3]=3

CauchyDistribution is a stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-zkvczc

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-d6bsov

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-bum801

Out[3]=3

NormalDistribution is a stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-xkrde7

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-49eymo

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-nuxytu

Out[3]=3

LevyDistribution is a stable distribution:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-yulxbd

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0bnhgp071je6i-fgdfgv

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0bnhgp071je6i-wo7n9z

Out[3]=3

Neat Examples (1)Surprising or curious use cases

PDFs for different β values with CDF contours:

In[1]:=1

✖

https://wolfram.com/xid/0bnhgp071je6i-io0104

In[4]:=4

✖

https://wolfram.com/xid/0bnhgp071je6i-ceikus

Out[4]=4

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StableDistribution

✖
`StableDistribution`

Details

Background & Context

Examples

Basic Examples (4)Summary of the most common use cases

Scope (4)Survey of the scope of standard use cases

Applications (9)Sample problems that can be solved with this function

Properties & Relations (11)Properties of the function, and connections to other functions

Neat Examples (1)Surprising or curious use cases

Text

CMS

APA

BibTeX

BibLaTeX

StableDistribution ✖ StableDistribution

Details

Background & Context

Examples

Basic Examples (4)Summary of the most common use cases

Scope (4)Survey of the scope of standard use cases

Applications (9)Sample problems that can be solved with this function

Properties & Relations (11)Properties of the function, and connections to other functions

Neat Examples (1)Surprising or curious use cases

See Also

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX

StableDistribution

✖
`StableDistribution`