Sinh-arcsinh normal distribution

class ctx.dist_sasnormal(a, b)

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

The sinh-arcsinh normal distribution distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), and the support interval \((0, +\infty)\).

See also: Jones and Faddy [413].

dist_sasnormal.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[\text{pdf}_X(x) = \frac{1}{\sqrt{2\pi}} \frac{\delta C_{\epsilon, \delta}(x)}{\sqrt{1+x^2}} \exp\left(-\tfrac{1}{2} S^2_{\epsilon, \delta}(x) \right), \quad \text{where } S_{\epsilon, \delta}(x) = \sinh\left(\epsilon + \delta \sinh^{-1}(x) \right), \text{and } C_{\epsilon, \delta}(x) = \sqrt{1+ S^2_{\epsilon, \delta}(x)}.\]

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

dist_sasnormal.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[\text{cdf}_X(x) = \Phi\left(S_{\epsilon, \delta}(x)\right), \quad \text{where } S_{\epsilon, \delta}(x) = \sinh\left(\epsilon + \delta \sinh^{-1}(x) \right).\]

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

dist_sasnormal.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[\text{sf}_X(x) = 1-\Phi\left(S_{\epsilon, \delta}(x)\right), \quad \text{where } S_{\epsilon, \delta}(x) = \sinh\left(\epsilon + \delta \sinh^{-1}(x) \right).\]

>>> 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_sasnormal.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[\text{qtf}_X(q) = S_{-\epsilon/\delta, 1/\delta} \left(\Phi^{-1}(q) \right), \quad \text{where } S_{\epsilon, \delta}(x) = \sinh\left(\epsilon + \delta \sinh^{-1}(x) \right).\]

>>> 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_sasnormal.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[\text{isf}_X(q) = S_{-\epsilon/\delta, 1/\delta} \left(\Phi^{-1}(1-q) \right), \quad \text{where } S_{\epsilon, \delta}(x) = \sinh\left(\epsilon + \delta \sinh^{-1}(x) \right).\]

>>> 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_sasnormal.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a sinh-arcsinh normal distribution 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_sasnormal.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[M_X(t) = ??\]

>>> 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_sasnormal.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a sinh-arcsinh normal distribution distribution:

\[K_X(t) = ??\]

>>> 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_sasnormal.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a sinh-arcsinh normal distribution distribution. All moments exist and are given by

\[\mu'_X(r) = \frac{1}{2^r} \sum_{j=0}^r (-1)^j \exp\left((r-2j)(\epsilon/\delta) \right) P_{(r-2j)/\delta}, \quad \text{where } P_q = \frac{e^{1/4}}{\sqrt{8\pi}} \left(K_{(q+1)/2}(1/4) + K_{(q-1)/2}(1/4) \right)\]

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

dist_sasnormal.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a sinh-arcsinh normal distribution 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