Halphen IB distribution

Halphen IB distribution#

class ctx.dist_halphen_ib(a, b)#

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

The Halphen IB (Inverse Halphen B) distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), and the support interval \((0, +\infty)\). See also Wikipedia [1243],

See also: Perreault et al. [490], Perreault et al. [491].

dist_halphen_ib.pdf(x)#

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

\[\text{pdf}_X(x) = \frac{2}{m^{-2\nu}N_{\nu}(\alpha)} x^{-2\nu-1} \exp \left[ -\left(\frac{m}{x}\right)^2 + \alpha \left(\frac{m}{x} \right) \right], x>0,\quad \text{where }\]
\[N_{\alpha} = \Gamma\left(\nu\right) \cdot {}_1F_1\left(\nu, \frac{1}{2}, \frac{\alpha^2}{4}\right) + \Gamma\left(\nu+\frac{1}{2}\right) \cdot {}_1F_1\left(\nu+\frac{1}{2}, \frac{3}{2}, \frac{\alpha^2}{4}\right)\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", fisher_f(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_halphen_ib.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Halphen IB 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_halphen_ib.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a Halphen IB 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_halphen_ib.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Halphen IB 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_halphen_ib.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a Halphen IB 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_halphen_ib.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Halphen IB 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_halphen_ib.m_x(t)#

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

dist_halphen_ib.k_x(t, k=0)#

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

dist_halphen_ib.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Halphen IB distribution. The rth moments only exists for \(r < 2\nu\).

\[\mu'_X(r) = m^r \frac{N_{\nu-r/2}(\alpha)}{N_{\nu}(\alpha)}.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_halphen_ib.cumulants(k)#

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