Gompertz-Makeham distribution#

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

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

Gompertz-Makeham distribution is a continuous probability distribution with parameters \(a > 0, b > 0\), \(\lambda \ge 0\), and the support interval \((0, +\infty)\). The hazard function has the form \(\text{hf}_X(x) = a e^{bx} + \lambda\).

See also: Wikipedia [1299], MathWorld [255], Jodrá [404], Riffi [504].

dist_gompertz.pdf(x)#

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

\[\text{pdf}_X(x) = \text{hf}_X(x) \cdot \text{sf}_X(x) = (a e^{bx} + \lambda) \cdot \exp \left[-\lambda x -\frac{a}{b} \left( e^{bx} - 1 \right) \right]\]

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

dist_gompertz.cdf(x)#

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

\[\text{cdf}_X(x) = 1 - \exp \left[-\lambda x -\frac{a}{b} \left( e^{bx} - 1 \right) \right] = -\mathrm{expm1}\left[-\lambda x -\frac{a}{b} \left(\mathrm{expm1}(bx) \right) \right]\]

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

dist_gompertz.sf(x)#

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

\[\text{sf}_X(x) = \exp \left[-\lambda x -\frac{a}{b} \left( e^{bx} - 1 \right) \right] = \exp \left[-\lambda x -\frac{a}{b} \left(\mathrm{expm1}(bx) \right) \right]\]

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

dist_gompertz.qtf(q)#

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

\[\begin{split}\text{qtf}_X(q) = {\begin{cases} \dfrac{1}{b} \mathrm{log1p}\left(-\dfrac{b}{a} \: \mathrm{log1p}(-q) \right) & {\text{for }}\lambda = 0,\\ \dfrac{a}{b \lambda} - \dfrac{1}{\lambda} \mathrm{log1p}(-q) - \dfrac{1}{b} W_0\left( \dfrac{a}{\lambda} e^{a/\lambda} \cdot \mathrm{pow1p} (-q, -b/\lambda) \right) & {\text{for }} \lambda > 0, \end{cases}}\end{split}\]

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_gompertz(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_gompertz.isf(q)#

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

\[\begin{split}\text{isf}_X(q) = {\begin{cases} \dfrac{1}{b} \mathrm{log1p}\left(-\dfrac{b}{a} \log(q) \right) & {\text{for }}\lambda = 0,\\ \dfrac{a}{b \lambda} - \dfrac{1}{\lambda} \log(q) - \dfrac{1}{b} W_0\left( \dfrac{a}{\lambda} e^{a/\lambda} \cdot q^{-b/\lambda} \right) & {\text{for }} \lambda > 0, \end{cases}}\end{split}\]

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_gompertz(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_gompertz.c_x(t)#

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

\[C_X(t) = e^d \left(d \cdot E_{p(it)}(d) + \frac{\lambda}{b} E_{1+p(it)}(d) \right), \quad \text{where } p(it) = \frac{\lambda-it}{b} \text{and } d=\frac{a}{b}.\]

Here \(E_p(z)\) denotes the exponential integral, defined as \(E_p(z) = z^{p-1} \Gamma(1-p,z) = z^{p-1} e^{-z} U(p,p,z)\). \(\Gamma(\cdot)\) denotes the incomplete gamma function (see …), and \(U(\cdot)\) denotes Tricomi’s hypergeometric function (see …).

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

dist_gompertz.m_x(t)#

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

\[M_X(t) = e^d \left(d \cdot E_{p(t)}(d) + \frac{\lambda}{b} E_{1+p(t)}(d) \right), \quad \text{where } p(t) = \frac{\lambda-t}{b} \text{and } d=\frac{a}{b}.\]

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

dist_gompertz.k_x(t, k=0)#

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Gompertz-Makeham 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_gompertz(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_gompertz.moments(k)#

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

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

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

dist_gompertz.cumulants(k)#

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

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

Gompertz-Makeham distribution

Contents

Gompertz-Makeham distribution#