Levy skew alpha-stable distribution

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Levy skew alpha-stable
Probability density function
Symmetric centered Lévy distributions
Symmetric centered Lévy distributions with unit scale factor
Skewed centered Lévy distributions
Skewed centered Lévy distributions with unit scale factor
Cumulative distribution function
CDF's for symmetric centered Lévy distributions
CDFs for symmetric centered Lévy distributions
CDF's for skewed centered Lévy distributions
CDFs for skewed centered Levy distributions
Parameters \alpha\in (0,2]\, exponent (real)

\beta\in [-1,1]\, skewness (real)
c\in [0,\infty)\, scale (real)
\mu \in (-\infty,\infty)\, location (real)
Support x \in (-\infty, +\infty)\! (real)
Probability density function (pdf) usually not analytically expressible (see text)
Cumulative distribution function (cdf) usually not analytically expressible (see text)
Mean undefined when α≤1, otherwise μ
Median usually not analytically expressible (see text). Equal to μ when β=0
Mode usually not analytically expressible. Equal to μ when β=0
Variance infinite except when α=2, when it is 2c2
Skewness undefined except when α=2, when it is 0
Excess kurtosis undefined except when α=2, when it is 0
Entropy not analytically expressible (see text)
Moment-generating function (mgf) undefined
Characteristic function \exp\left[~it\mu - |c t|^\alpha\,(1-i \beta\,\mbox{sgn}(t)\Phi)~\right]

\Phi=\tan(\pi \alpha/2)\, for \alpha \ne 1\,
\Phi=-(2/\pi)\log|t|\, for \alpha = 1\,

In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a family of continuous probability distributions which are characterized by four parameters: α, β, μ and c , as well as the distributed value, x . The μ and c  are shift and scale parameters resepectively, while - loosely speaking - the β parameter controls the skewness and the α parameter controls the kurtosis. The stable distribution has the important property of stability: If a number of independent identically distributed (iid) random variables have a stable distribution, then a linear combination of these variables will have the same distribution, except for possibly different shift and scale parameters. To be more precise:

If X1 and X2 are distributed according to a stable distribution L(x;α,β,μ,c), and if Y = AX1 + BX2 + C is a linear combination of the two, then there exist values of D  and E  such that DY + E is distributed according to a stable distribution L(DY + E;α,β,μ,c) or, equivalently, Y is distributed according to a stable distribution L(Y;α,β,(μ − E) / D,c / D). If E = 0 for all A, B, and C then Y is said to have a strictly stable distribution. Since the normal distribution, the Cauchy distribution, and the Lévy distribution all have the above property, it follows that they are special cases of the stable distribution (Nolan 2005).

Stable distributions owe their importance in both theory and practice to the generalization of the Central Limit Theorem to random variables without second (and possibly first) order moments and the accompanying self-similarity of the stable family. It was the seeming departure from normality along with the demand for a self-similar model for financial data (i.e. the shape of the distribution for yearly asset price changes should resemble that of the constituent daily or monthly price changes) that led Benoît Mandelbrot to propose that cotton prices follow a Lévy skew alpha-stable distribution with α equal to 1.7. Levy skew alpha-stable distributions are frequently found in analysis of critical behavior and financial data. (Voit 2003 § 5.4.3) Lévy skew alpha-stable distributions are also found in spectroscopy as a general expression for a quasistatically pressure-broadened spectral line. (Peach 1981 § 4.5) All stable distributions are infinitely divisible and with the exception of the normal distribution for which α=2, stable distributions are leptokurtotic and Heavy-tailed distributions.

Contents

[edit] The distribution

A Lévy skew stable distribution is specified by scale c, exponent α, shift μ and skewness parameter β. The skewness parameter must lie in the range [−1, 1] and when it is zero, the distribution is symmetric and is referred to as a Lévy symmetric alpha-stable distribution. The exponent α must lie in the range (0, 2].

The Lévy skew stable probability distribution is defined by the Fourier transform of its characteristic function \varphi(t) (Voit 2003 § 5.4.3)(Nolan 2005)

f(x;\alpha,\beta,c,\mu) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} \varphi (t) e^{-itx}\,dt

where \varphi(t) is given by:

\varphi(t) = \exp\left[~it\mu\!-\!|c t|^\alpha\,(1\!-\!i \beta\,\textrm{sgn}(t)\Phi)~\right]

where sgn(t)  is just the sign of t and Φ is given by

\Phi=\tan(\pi \alpha/2)\,

for all α except α = 1 in which case:

\Phi=-(2/\pi)\log|t|.\,

μ is a shift parameter, β is a measure of asymmetry, with β=0 yielding a distribution symmetric about μ. c is a scale factor which is a measure of the width of the distribution and α is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution for α < 2. Note that this is only one of the parameterizations in use for stable distributions; it is the most common but is not continuous in the parameters.

The asymptotic behavior is described, for α<2, by: (Nolan, Theorem 1.12)

f(x)\sim\frac{\alpha c^\alpha (1+\beta) \sin(\pi \alpha / 2)\Gamma(\alpha)/\pi}{|x|^{1+\alpha}}

where Γ is the Gamma function (except that when α<1 and β=1 or -1, the tail vanishes to the left or right, resp., of μ). This "heavy tail" behavior causes the variance of Lévy distributions to be infinite for all α < 2. This property is illustrated in the log-log plots below.

Log-log plot of symmetric centered Levy distribution PDF's showing the power law behavior for large x. The power law behavior is evidenced by the straight-line appearance of the PDF for large x, with the slope equal to -(α+1). (The only exception is for α=2, in black, which is a normal distribution.)
Log-log plot of symmetric centered Levy distribution PDF's showing the power law behavior for large x. The power law behavior is evidenced by the straight-line appearance of the PDF for large x, with the slope equal to -(α+1). (The only exception is for α=2, in black, which is a normal distribution.)
Log-log plot of skewed centered Levy distribution PDF's showing the power law behavior for large x. Again the slope of the linear portions is equal to -(α+1)
Log-log plot of skewed centered Levy distribution PDF's showing the power law behavior for large x. Again the slope of the linear portions is equal to -(α+1)


When α=2, the distribution is Gaussian (see below), with tails asymptotic to exp(-x2/4c2)/(2c√π).

[edit] Special cases

There is no general analytic solution for the form of p(x). There are, however three special cases which can be analytically expressed as can be seen by inspection of the characteristic function.

Note that the above three distributions are also connected, in the following way: A standard Cauchy random variable can be viewed as a mixture of Gaussian random variables (all with mean zero), with the variance being drawn from a standard Lévy distribution. And in fact this is a special case of a more general theorem which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to unity).

Other special cases are:

  • In the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function δ(x − μ).

An interactive tutorial of Stable Laws may be found at http://www.mathestate.com/tools/Financial/map/Overview.html

[edit] Stability property

(See (Voit 2003 § 5.4.3) and (Nolan 2005))

The Lévy alpha-stable distributions have the "stability" property that if N alpha-stable variates Xi are drawn from the distribution

X_i \sim f(x;\alpha, \beta,c,\mu)\,

then the sum

Y = \sum_{i=1}^N k_i (X_i-\mu)\,

will also be distributed as an alpha-stable variate,

Y \sim \frac{1}{s}\,\,f(y/s;\alpha,\beta,c,0).\,

where

s=\left(\sum_{i=1}^N |k_i|^\alpha\right)^{1/\alpha}.\,

This can be easily proven using the properties of characteristic functions.

[edit] The generalized central limit theorem

Another important property of Lévy distributions is the role that they play in a generalized central limit theorem. The central limit theorem states that the sum of a number of random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | α + 1 (and therefore having infinite variance) will tend to a stable Levy distribution f(x;α,0,c,0) as the number of variables grows. (Voit 2003 § 5.4.3)

[edit] Series representation

The stable distribution can be restated as the real part of simpler integral:(Peach 1981 § 4.5)

f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \int_0^\infty e^{it(x-\mu)}e^{-(ct)^\alpha(1-i\beta\Phi)}\,dt\right]

Expressing the second exponential as a Taylor series, we have:

f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \int_0^\infty e^{it(x-\mu)}\sum_{n=0}^\infty\frac{(-qt^\alpha)^n}{n!}\,dt\right]

where q = cα(1 − iβΦ). Reversing the order of integration and summation, and carrying out the integration yields:

f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[ \sum_{n=1}^\infty\frac{(-q)^n}{n!}\left(\frac{i}{x-\mu}\right)^{\alpha n+1}\Gamma(\alpha n+1)\right]

which will be valid for x\ne\mu and will converge for appropriate values of the parameters. (Note that the n=0 term which yields a delta function in x − μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x − μ which is generally less useful.

[edit] See also

[edit] References


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Probability distributions
 
Continuous univariate supported on the whole real line