Feller-Pareto distribution

class ctx.dist_feller_pareto(a, b)

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

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

The Feller-Pareto distribution is defined as the distribution of a random variable \(X\) with

\[X = \mu + \theta \left( \frac{U}{V} \right)^{1/\gamma}, \quad \gamma, \theta>0, \mu \in \mathbb{R},\]

where \(U\) and \(V\) are two independent gamma distributions with shape parameter \(\tau>0\) and \(\alpha>0\), respectively, and common scale parameter 1.

See also: Wikipedia [1294], Dutang et al. [307].

For \(\mu = \theta, \gamma = \tau = 1\) we obtain the Pareto Type I distribution.

For \(\gamma = \tau = 1\) we obtain the Pareto Type II distribution. When \(\mu = 0\), we obtain what is generally simply called the Pareto distribution.

For \(\alpha = \tau = 1\), we obtain the Pareto Type III distribution.

For \(\tau = 1\) we obtain the Pareto Type IV distribution.

For \(\mu=0\), this reduces to the four parameter generalized beta of the second kind distribution (see GB2), with \(\text{pdf}_X(x) = f(a,b,p,q)\) (McDonald 1984), where \(a=\theta, b=\gamma, p=\tau, q=\alpha\).

dist_feller_pareto.pdf(x)

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

\[\text{pdf}_X(x) = \frac{\gamma \cdot ((x-\mu)/\theta)^{\gamma \tau-1}}{\theta B(\alpha, \tau) [1 + ((x-\mu)/\theta)^{\gamma} ]^{\alpha+\tau} } = \frac{\gamma u^{\tau}(1-u)^{\alpha}}{(x-\mu)B(\alpha, \tau)}, \quad \text{where } u = \frac{y}{1+y}, y = \left( \frac{x-\mu}{\theta} \right)^{\gamma}, \quad x \ge \mu.\]

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

dist_feller_pareto.cdf(x)

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

\[\text{cdf}_X(x) = I(\tau, \alpha; u), \quad \text{where } u = \frac{y}{1+y}, y = \left( \frac{x-\mu}{\theta} \right)^{\gamma}, \quad x \ge \mu.\]

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

dist_feller_pareto.sf(x)

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

\[\text{sf}_X(x) = 1 - I(\tau, \alpha; u), \quad \text{where } u = \frac{y}{1+y}, y = \left( \frac{x-\mu}{\theta} \right)^{\gamma}, \quad x \ge \mu.\]

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_feller_pareto.qtf(q)

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

\[\text{qtf}_X(q) = ??\]

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_feller_pareto.isf(q)

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

\[\text{isf}_X(q) = ??\]

Here \(\mathrm{ibetac\_inv}(\cdot)\) denotes the inverse of the real normalised complementary 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_feller_pareto.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Feller-Pareto 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_feller_pareto.m_x(t)

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

dist_feller_pareto.k_x(t, k=0)

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

dist_feller_pareto.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Feller-Pareto distribution. The kth moments only exists for \(k < \alpha \gamma\).

\[\mu'_X(k) = \sum_{j=0}^k \binom{k}{j} \frac{\mu^{k-j} \theta^j \Gamma(\tau+j/\gamma) \Gamma(\alpha-j/\gamma)}{\Gamma(\alpha) \Gamma(\tau)}\]

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

dist_feller_pareto.cumulants(k)

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