Dagum (Burr Type III) distribution

Dagum (Burr Type III) distribution#

class ctx.dist_dagum(n1, n2, lambda, **kwargs)#

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

These functions return PDF, CDF, and ICDF of the Dagum distribution with location \(a > 0\), scale \(b > 0\), and the support interval \((0,+\infty)\).

See also: Wikipedia [1290], Kleiber and Kotz [415], Dagum [201].

dist_dagum.pdf(x)#

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

\[\text{pdf}_X(x) = f(x;a,b,p)={\frac {ap}{x}} {\frac {(x/b)^{ap}}{\left((x/b)^{a}+1\right)^{p+1}}}.\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", dist_dagum(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_dagum.cdf(x)#

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

\[\text{cdf}_X(x) = \left[ 1+\left(\frac{x}{b}\right)^{-a} \right]^{-p} \quad \text{for } x>0, \text{ where } a,b,p >0\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", dist_dagum(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_dagum.sf(x)#

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

\[\text{sf}_X(x) = 1 - \left[ 1+\left(\frac{x}{b}\right)^{-a} \right]^{-p}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", dist_dagum(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_dagum.qtf(q)#

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

\[\text{qtf}_X(q) = b(q^{-1/p}-1)^{-1/a}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", dist_dagum(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_dagum.isf(q)#

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

\[\text{isf}_X(q) = b((1-q)^{-1/p}-1)^{-1/a}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", dist_dagum(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_dagum.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an Dagum 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_dagum(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_dagum.m_x(t)#

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

dist_dagum.k_x(t, k=0)#

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

dist_dagum.moments(k)#

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Dagum distribution (see Kleiber_2007_Dagum_moments). The kth moment exists for \(-ap < k < a\) and equals

\[\mu_k = \frac{b^k \Gamma(p+k/a) \Gamma(1-k/a)}{\Gamma(p)}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", dist_dagum(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_dagum.cumulants(k)#

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

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