Generalized hyperbolic distribution

Generalized hyperbolic distribution#

class ctx.dist_gen_hyperbolic(a, b)#

where ctx is dec, mpm, ipm, fpm, gmp or arb.

The generalized hyperbolic distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), and the support interval \((0, +\infty)\).

See also Wikipedia [1317],

dist_gen_hyperbolic.pdf(x)#

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

\[\text{pdf}_X(x) = \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)} \; e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}}\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", fisher_f(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_gen_hyperbolic.cdf(x)#

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

\[\text{cdf}_X(x) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", fisher_f(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_gen_hyperbolic.sf(x)#

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

\[\text{sf}_X(x) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", fisher_f(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_gen_hyperbolic.qtf(q)#

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

\[\text{qtf}_X(q) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", fisher_f(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_gen_hyperbolic.isf(q)#

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

\[\text{isf}_X(q) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", fisher_f(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_gen_hyperbolic.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a generalized hyperbolic distribution:

\[C_X(t) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", fisher_f(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_gen_hyperbolic.m_x(t)#

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a generalized hyperbolic distribution:

\[M_X(t) = \frac{e^{\mu t}\gamma^\lambda}{(\sqrt{\alpha^2 -(\beta +t)^2})^\lambda} \frac{K_\lambda(\delta \sqrt{\alpha^2 -(\beta +t)^2})}{K_\lambda (\delta \gamma)}\]

dist_gen_hyperbolic.k_x(t, k=0)#

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a generalized hyperbolic distribution:

\[K_X(t) = ??\]

dist_gen_hyperbolic.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a generalized hyperbolic distribution. The rth moments only exists for \(n_2 > 2r\).

\[\mu'_X(r) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_gen_hyperbolic.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a generalized hyperbolic distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00
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