Fisk (log-logistic) distribution

class ctx.dist_fisk(n1, n2, lambda, **kwargs)

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

The Fisk distribution is a continuous probability distribution with scale parameter \(a > 0\), shape parameter \(b > 0\), and the support interval \([0, +\infty)\).

See also: Wikipedia [1306], Kleiber and Kotz [415].

dist_fisk.pdf(x)

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

\[\text{pdf}_X(x) = f(x;\alpha ,\beta )={\frac {(\beta /\alpha )(x/\alpha )^{{\beta -1}}}{\left(1+(x/\alpha )^{{\beta }}\right)^{2}}}\]

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

dist_fisk.cdf(x)

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

\[\text{cdf}_X(x) = \frac{1}{1+(x/\alpha)^{-\beta}} = \frac{x^{\beta}}{\alpha^{\beta} + x^{\beta}}\]

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

dist_fisk.sf(x)

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

\[\text{sf}_X(x) = 1-\frac{1}{1+(x/\alpha)^{-\beta}} = 1-\frac{x^{\beta}}{\alpha^{\beta} + x^{\beta}}\]

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

dist_fisk.qtf(q)

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

\[\text{qtf}_X(q) = \alpha \left( \frac{p}{1-p} \right) ^{1/\beta}\]

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

dist_fisk.isf(q)

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

\[\text{isf}_X(q) = \alpha \left( \frac{q}{1-q} \right) ^{1/\beta}\]

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

dist_fisk.c_x(t)

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

\[C_X(t) = \int_{0}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]

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

dist_fisk.m_x(t)

Returns None, since the moment generating function does not exist.

dist_fisk.k_x(t, k=0)

Returns None, since the cumulant generating function does not exist.

dist_fisk.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Fisk distribution (see Kleiber_2007_Dagum_moments). The kth raw moment exists only when \(k<\beta\) , when it is given by

\[\mu_k = \alpha^k \frac{k \pi/\beta}{\sin(k \pi/\beta)}\]

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

dist_fisk.cumulants(k)

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

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