Skew exponential power distribution

class ctx.skewexponpower(a, b)

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

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

See also: Hutson [395], Kleiber and Kotz [415] (p. 131), Wikipedia [1303], MathWorld [254].

skewexponpower.pdf(x)

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

\[\text{pdf}_X(x) = \frac{1}{k\sigma} \exp \left( -\tfrac{1}{2}\left( |z|+(2\alpha-1)z \right) \right)^{1/c}, \quad \text{where } c=\frac{1+\beta}{2}, \quad k=\frac{4\alpha(1-\alpha)}{\Gamma(1+c) 2^{1+c}}, \quad z=\frac{x-\theta}{\sigma}.\]

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

skewexponpower.cdf(x)

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

\[\begin{split}\text{cdf}_X(x) = \begin{cases} \alpha P\left(c, 2^{(1/c)-1} ((\alpha-1)z)^{1/c} \right) & \text{if } z \le 0 \\ 1-(1-\alpha) P\left(c, 2^{(1/c)-1} (\alpha z)^{1/c} \right) & \text{if } z > 0 \end{cases} , \quad \text{where } c=\frac{1+\beta}{2}, \quad z=\frac{x-\theta}{\sigma}.\end{split}\]

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

skewexponpower.sf(x)

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

\[\begin{split}\text{sf}_X(x) = \begin{cases} 1-\alpha P\left(c, 2^{(1/c)-1} ((\alpha-1)z)^{1/c} \right) & \text{if } z \le 0 \\ (1-\alpha) P\left(c, 2^{(1/c)-1} (\alpha z)^{1/c} \right) & \text{if } z > 0 \end{cases} , \quad \text{where } c=\frac{1+\beta}{2}, \quad z=\frac{x-\theta}{\sigma}.\end{split}\]

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

skewexponpower.qtf(q)

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

\[\begin{split}\text{qtf}_X(q) = \begin{cases} \theta + \frac{\sigma}{1-\alpha} \left( 2^{((1/c)-1)/2} \sqrt{ P^{-1}\left(c, q/\alpha \right)} \right)^{\beta+1} & \text{if } q \le \alpha \\ \theta + \frac{\sigma}{\alpha} \left( 2^{((1/c)-1)/2} \sqrt{ P^{-1}\left(c, (1-q)/(1-\alpha) \right)} \right)^{\beta+1} & \text{if } q > \alpha \end{cases} , \quad \text{where } c=\frac{1+\beta}{2}.\end{split}\]

>>> 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

skewexponpower.isf(q)

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

\[\begin{split}\text{isf}_X(q) = \begin{cases} \theta + \frac{\sigma}{1-\alpha} \left( 2^{((1/c)-1)/2} \sqrt{ P^{-1}\left(c, q/\alpha \right)} \right)^{\beta+1} & \text{if } q \le \alpha \\ \theta + \frac{\sigma}{\alpha} \left( 2^{((1/c)-1)/2} \sqrt{ P^{-1}\left(c, (1-q)/(1-\alpha) \right)} \right)^{\beta+1} & \text{if } q > \alpha \end{cases} , \quad \text{where } c=\frac{1+\beta}{2}.\end{split}\]

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

skewexponpower.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a skew exponential power 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

skewexponpower.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a skew exponential power 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

skewexponpower.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a skew exponential power 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

skewexponpower.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a skew exponential power distribution. All moments exist. The moments of \(Z = (X-\theta)/\sigma\) are given by

\[\mu'_Z(r) = \frac{(\beta+1) \cdot \left( (-2)^r \alpha^{r+1} + (1-\alpha)(2(1-\alpha))^r \right) \cdot 2^{(\beta-3)r/2-1} \cdot \Gamma((\beta+1)(r+1)/2)}{(1-\alpha)^r \alpha^r \Gamma((\beta+3)/2)}\]

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

skewexponpower.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a skew exponential power 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