Fréchet (Maximum/Minimum-Type-II or Inverse Weibull) distribution

Fréchet (Maximum/Minimum-Type-II or Inverse Weibull) distribution#

class ctx.dist_frechet(a, b=1)#

The Fréchet distribution is a continuous probability distribution with shape parameters \(a\) and \(b\), and the support interval \((0,+\infty)\). If \(b = 1\) this is the classical Fréchet distribution. For \(b \ne 1\) this is also called reciprocal Weibull distribution or the generalized inverse Weibull distribution ([DeGusmao2011`).

See also: Wikipedia [1296], De Gusmao et al. [212], Kleiber and Kotz [415],

TODO: remove m and s from these formulas, use formulas from De Gusmao et al. [212].

Note: \(\alpha\) in Wikipedia corresponds to \(-\beta\) in De Gusmao et al. [212].

dist_frechet.pdf(x)#

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

\[\text{pdf}_X(x) = \frac{\alpha}{s} \; \left(\frac{x-m}{s}\right)^{-1-\alpha} \; e^{-(\frac{x-m}{s})^{-\alpha}}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", dist_frechet(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_frechet.cdf(x)#

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

\[\text{cdf}_X(x) = e^{-(\frac{x-m}{s})^{-\alpha}}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", dist_frechet(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_frechet.sf(x)#

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

\[\text{sf}_X(x) = 1 - e^{-(\frac{x-m}{s})^{-\alpha}}\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", dist_frechet(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_frechet.qtf(q)#

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

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

dist_frechet.isf(q)#

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

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

dist_frechet.c_x(t)#

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

dist_frechet.m_x(t)#

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

dist_frechet.k_x(t, k=0)#

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

dist_frechet.moments(k)#

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Fréchet distribution. Moments exist only for \(a > k\).

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

dist_frechet.cumulants(k)#

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

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