Lévy alpha-stable distribution

class ctx.dist_levy_alpha_stable(n1, n2, lambda, **kwargs)

A random variable \(X\) is called Lévy alpha-stable if its characteristic function, \(C_X(t)\), can be written as

\[C_X(t) = \varphi (t;\alpha ,\beta ,c,\mu )=\exp \left(it\mu -|ct|^{\alpha }\left(1-i\beta \operatorname {sgn} (t)\Phi \right)\right)\]

where \(sgn(t)\) is just the sign of \(t\) and

\[\begin{split}\Phi ={\begin{cases} \tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |t|&\alpha =1 \end{cases}}\end{split}\]

where \(\mu \in \mathbb{R}\) is a shift parameter.

See also: Wikipedia [1330], Witkovský [1647].

dist_levy_alpha_stable.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a Lévy alpha stable distribution:

\[\text{pdf}_X(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ity} C_X(t) \mathrm{d} y = \frac{1}{\pi} \int_{0}^{\infty} \Re \left ( e^{-itx} C_X(t) \right ) \mathrm{d} t.\]

where \(\Re (z)\) denotes the real part of \(z\).

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", dist_levy_alpha_stable(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_levy_alpha_stable.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Lévy alpha stable distribution:

\[\text{cdf}_X(x) = \frac{1}{2} - \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{e^{-itx} C_X(t) - e^{itx} C_X(t)}{it} \mathrm{d} t = \frac{1}{2} - \frac{1}{\pi} \int_{0}^{\infty} \Im \left ( \frac{ e^{-itx} C_X(t)}{t} \right ) \mathrm{d} t.\]

where \(\Im (z)\) denotes the imaginary part of \(z\).

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", dist_levy_alpha_stable(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_levy_alpha_stable.sf(x)

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following a Lévy alpha stable distribution:

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", dist_levy_alpha_stable(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_levy_alpha_stable.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following a Lévy alpha stable distribution:

There is no known explicit form for the quantile function \(\text{cdf}^{-1}_X(x)\): It is computed using Newton iterations with starting values from a normal approximation.

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", dist_levy_alpha_stable(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_levy_alpha_stable.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following a Lévy alpha stable distribution:

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", dist_levy_alpha_stable(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_levy_alpha_stable.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Lévy alpha stable distribution. The characteristic function can be written as

\[C_X(t) = \varphi (t;\alpha ,\beta ,c,\mu )=\exp \left(it\mu -|ct|^{\alpha }\left(1-i\beta \operatorname {sgn} (t)\Phi \right)\right)\]

where sgn`(t)` is just the sign of \(t\) and

\[\begin{split}\Phi ={\begin{cases} \tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\-{\frac {2}{\pi }}\log |t|&\alpha =1 \end{cases}}\end{split}\]

where \(\mu \in \mathbb{R}\) is a shift parameter.

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", dist_levy_alpha_stable(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_levy_alpha_stable.m_x(t)

Returns NaN, since the moment generating function does not exist.

dist_levy_alpha_stable.k_x(t, k=0)

Returns NaN, since the cumulant generating function does not exist.

dist_levy_alpha_stable.moments(k)

Returns NaN, since moments do not exist.

dist_levy_alpha_stable.cumulants(k)

Returns NaN, since cumulants do not exist.

Examples

A number of cases of analytically expressible stable distributions are known. Let the stable distribution be expressed by \(f(x;\alpha ,\beta ,c,\mu )\), then we know:

The Cauchy Distribution is given by \(f(x;\alpha=1,\beta=0,c=1,\mu=0)\).

The Levy distribution is given by \(f(x;\alpha={\tfrac {1}{2}},\beta=1,c=1,\mu=0)\).

The Normal distribution is given by \(f(x;\alpha=2,\beta=0,c=1,0)\).

A special case is the Landau distribution, which is given by \(f(x;\alpha=1,\beta=1,c=\pi/2,\mu=0)\).

Let \(S_{\mu ,\nu }(z)\) be a Lommel function, then

\[f\left( x;\alpha={\tfrac {1}{3}},\beta=0,c=1,\mu=0 \right) = \Re \left({\frac {2e^{-{\frac {i\pi }{4}}}}{3{\sqrt {3}}\pi }}{\frac {1}{\sqrt {x^{3}}}}S_{0,{\frac {1}{3}}}\left({\frac {2e^{\frac {i\pi }{4}}}{3{\sqrt {3}}}}{\frac {1}{\sqrt {x}}}\right)\right)\]

Let \(S(x)\) and \(C(x)\) denote the Fresnel Integrals then:

\[f\left(x;\alpha={\tfrac {1}{2}},\beta=0,c=1,\mu=0\right) = {\frac {1}{\sqrt {2\pi |x|^{3}}}}\left(\sin \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-S\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]+\cos \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-C\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]\right).\]

Let \(K_{v}(x)\) be the modified Bessel function of the second kind then:

\[f\left(x;\alpha={\tfrac {1}{3}},\beta=1,c=1,\mu=0\right) = {\frac {1}{\pi }}{\frac {2{\sqrt {2}}}{3^{\frac {7}{4}}}}{\frac {1}{\sqrt {x^{3}}}}K_{\frac {1}{3}}\left({\frac {4{\sqrt {2}}}{3^{\frac {9}{4}}}}{\frac {1}{\sqrt {x}}}\right)\]