Boost: Skew normal Distribution

The following functions return the pdf, cdf, qtf or boost class of the skew normal distribution with location \(a \in \mathbb{R}\), scale \(b > 0\), shape \(c \in \mathbb{R}\), and the support interval \((-\infty, +\infty)\).

See also Wikipedia [1283], MathWorld [264], BoostMath [92], Haas [376].

Ctx.skewnormal_pdf(x, a, b, c)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the skew normal distribution:

\[\text{pdf}(x) = \frac{2}{b} \phi \left(\frac{x-a}{b}\right) \Phi \left(c \left(\frac{x-a}{b}\right)\right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("SkewnormalPdf(x, a, b): ", SkewnormalPdf(x, a, b))
>>> print ("dist_skewnormal(a, b).pdf(x): ", dist_skewnormal(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.skewnormal_cdf(x, a, b, c)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the skew normal distribution:

\[\text{cdf}(x) = \Phi \left(\frac{x-a}{b}\right) - 2T \left(\frac{x-a}{b}, c \right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("SkewnormalCdf(x, a, b): ", SkewnormalCdf(x, a, b))
>>> print ("dist_skewnormal(a, b).cdf(x): ", dist_skewnormal(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.skewnormal_qtf(q, a, b, c)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the skew normal distribution:

There is no known closed form for \(\text{qtf}(q)\): it computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("SkewnormalQtf(q, a, b): ", SkewnormalQtf(q, a, b))
>>> print ("dist_skewnormal(a, b).qtf(q): ", dist_skewnormal(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_skewnormal(a, b, c)

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

The skew normal distribution is a continuous probability distribution with location \(a \in \mathbb{R}\), scale \(b > 0\), shape \(c \in \mathbb{R}\), and the support interval \((-\infty, +\infty)\). See also Wikipedia [1283], MathWorld [264], BoostMath [92], Witkovský [1646], Haas [376].

dist_skewnormal.pdf(x)

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

\[\text{pdf}_X(x) = \frac{2}{b} \phi \left(\frac{x-a}{b}\right) \Phi \left(c \left(\frac{x-a}{b}\right)\right).\]

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

dist_skewnormal.cdf(x)

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

\[\text{cdf}_X(x) = \Phi \left(\frac{x-a}{b}\right) - 2T \left(\frac{x-a}{b}, c \right).\]

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

dist_skewnormal.sf(x)

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

\[\text{sf}_X(x) = \Phi \left(-\frac{x-a}{b}\right) + 2T \left(\frac{x-a}{b}, c \right).\]

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

dist_skewnormal.qtf(q)

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

There is no known closed form for \(\text{qtf}_X(q)\) or \(\text{isf}_X(q)\): These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).

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

dist_skewnormal.isf(q)

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

There is no known closed form for \(\text{qtf}_X(q)\) or \(\text{isf}_X(q)\): These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).

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

dist_skewnormal.c_x(t)

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

\[C_X(t) = 2 \exp \left( ita - \frac{b^2 t^2}{2} \right) \Phi(it bd).\]

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

dist_skewnormal.m_x(t)

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

\[M_X(t) = 2 \exp \left( ta + \frac{b^2 t^2}{2} \right) \Phi(t bd).\]

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

dist_skewnormal.k_x(t, k=0)

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

\[K_X(t) = ta + \frac{b^2 t^2}{2} + \log \left( 2 \Phi(t bd) \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", mp_skewnormal(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_skewnormal.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a skew normal distribution:

\[\mu'_{2r+1} = \sqrt{\frac{2}{\pi}} \frac{(2r+1)!}{2^r r!} \sum_{j=0}^{r} (-1)^j \binom{r}{j} \frac{d^{2j+1}}{2j+1}\]

The even moments are equal to those of the standard normal. Hass 2012: Odd Moments

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

dist_skewnormal.cumulants(k)

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

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