Skew t-distribution (Jones)

class ctx.dist_skewt(a, b)

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

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

See also: Jones and Faddy [412].

Other skew t-distributions: see downloads.

dist_skewt.pdf(x)

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

\[\text{pdf}_X(x) = \frac{1}{2^{a+b-1}B(a,b)\sqrt{a+b}} \times \left( 1 + \frac{x}{\sqrt{a+b+x^2}} \right)^{a+1/2} \times \left( 1 - \frac{x}{\sqrt{a+b+x^2}} \right)^{b+1/2}\]

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

dist_skewt.cdf(x)

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

\[\text{cdf}_X(x) = I_z(a,b), \quad \text{where } z = \frac{1}{2}\left( 1 + \frac{x}{\sqrt{a+b+x^2}} \right)\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function, and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function.

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

dist_skewt.sf(x)

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

\[\text{sf}_X(x) = 1-I_z(a,b), \quad \text{where } z = \frac{1}{2}\left( 1 + \frac{x}{\sqrt{a+b+x^2}} \right)\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function, and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function.

>>> 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_skewt.qtf(q)

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

\[\text{qtf}_X(q) = \frac{\sqrt{a+b}(2Y-1)}{2\sqrt{Y(1-Y)}},\]

where \(Y\) is the quantile of the beta distribution. Here \(\mathrm{ibeta\_inv}(\cdot)\) denotes the inverse of the real normalised incomplete beta function.

>>> 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_skewt.isf(q)

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

\[\text{isf}_X(q) = \frac{\sqrt{a+b}(2Y-1)}{2\sqrt{Y(1-Y)}},\]

where \(Y\) is the quantile of the beta distribution. Here \(\mathrm{ibeta\_inv}(\cdot)\) denotes the inverse of the real normalised incomplete beta function.

>>> 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_skewt.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a skew-t-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_skewt.m_x(t)

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

dist_skewt.k_x(t, k=0)

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

dist_skewt.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a skew-t-distribution. The rth moments only exists for \(a > r/2\) and \(b > r/2\).

\[\mu'_X(r) = \frac{(a+b)^{r/2}}{B(a,b)} \sum_{i=0}^r \binom{r}{i} 2^{-i} (-1)^i B\left(a+\frac{r}{2}-i, b-\frac{r}{2} \right)\]

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

dist_skewt.cumulants(k)

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