Singh-Maddala (Burr Type XII) distribution

Singh-Maddala (Burr Type XII) distribution#

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

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

See also: Wikipedia [1289], Rodriguez [508], Kleiber and Kotz [415] (page 198), Kumar [424],

dist_singh_maddala.pdf(x)#

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

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

dist_singh_maddala.cdf(x)#

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

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

dist_singh_maddala.sf(x)#

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

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

dist_singh_maddala.qtf(d)#

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

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

dist_singh_maddala.isf(d)#

Returns \(\text{isf}_X(d)\), the inverse survival function function (isf) of a random variable \(X\), following an Singh-Maddala distribution:

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

dist_singh_maddala.c_x(t)#

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

dist_singh_maddala.m_x(t)#

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

dist_singh_maddala.k_x(t, k=0)#

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

dist_singh_maddala.moments(k)#

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Singh-Maddala distribution. The kth moment exists for \(-a < k < aq\); it equals (see Kleiber(2003), page 201)

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

dist_singh_maddala.cumulants(k)#

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

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