Shifted Gompertz distribution

class ctx.dist_shifted_gompertz(a, b, lambda1=0, **kwargs)

Shifted Gompertz distribution is a continuous probability distribution with parameters \(a > 0, b > 0\), and the support interval \([0, +\infty)\).

See also: Wikipedia [1309], MathWorld [263], Jiménez and Pedro Jodrá [399], Jiménez Torres [400].

dist_shifted_gompertz.pdf(x)

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

\[\text{pdf}_X(x) = b e^{-(bx + a e^{-bx})} \left(1 + a \left(1 - e^{-bx} \right) \right), \quad x>0.\]

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

dist_shifted_gompertz.cdf(x)

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

\[\text{cdf}_X(x) = \left(1 - e^{-bx}\right) e^{-ae^{-bx}}, \quad x>0.\]

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

dist_shifted_gompertz.sf(x)

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

\[\text{sf}_X(x) = 1 - \left(1 - e^{-bx}\right) e^{-ae^{-bx}}, \quad x>0.\]

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

dist_shifted_gompertz.qtf(q)

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

\[\text{qtf}_X(q) = \frac{1}{b} \log \left(1 - \frac{W_0(a e^a q)}{a} \right)\]

where \(W_0\) denotes the principal branch of the Lambert \(W\) function (see …).

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

dist_shifted_gompertz.isf(q)

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

\[\text{isf}_X(q) = \frac{1}{b} \log \left(1 - \frac{W_0(a e^a (1-q))}{a} \right)\]

where \(W_0\) denotes the principal branch of the Lambert \(W\) function (see …).

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

dist_shifted_gompertz.c_x(t)

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

\[C_X(t) = a^{i t/(b-1)} (a + i t/b) \Gamma(1-i t/b, a) + e^{-a}.\]

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

dist_shifted_gompertz.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an Shifted Gompertz distribution:

\[M_X(t) = a^{t/(b-1)} (a + t/b) \Gamma(1-t/b, a) + e^{-a}.\]

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

dist_shifted_gompertz.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Shifted Gompertz distribution:

\[K_X(t) = \log(M_X(t))\]

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

dist_shifted_gompertz.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Shifted Gompertz distribution. The moments are calculated from their definition:

\[\mu'_X(r) = E(X^r) = \int_{0}^{1} x^r \text{pdf}_X(x) \mathrm{d} x\]

There is an explicit expression for the mean:

\[\mu'_X(1) = E(X) = \frac{1}{b} \left(\gamma + \log(a) + \frac{1-e^{-a}}{a} + E_1(a) \right).\]

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

dist_shifted_gompertz.cumulants(k)

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

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