Stacy (generalized gamma) distribution

class ctx.dist_stacy(a, b)

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

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

See also Wikipedia [1301], Crooks [194] (p.87).

dist_stacy.pdf(x)

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

\[\text{pdf}_X(x) = \frac{|\beta|}{\Gamma(\alpha) |\theta|} \left(\frac{x}{\theta} \right)^{\alpha \beta -1} \exp \left(-\left( \frac{x}{\theta} \right)^{\beta} \right)\]

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

dist_stacy.cdf(x)

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

\[\text{cdf}_X(x) = P(\alpha, z) \quad \text{for } \beta/\theta > 0, \quad \text{where } z = \left( \frac{x}{\theta} \right)^{\beta}\]

\[\text{cdf}_X(x) = Q(\alpha, z) \quad \text{for } \beta/\theta < 0, \quad \text{where } z = \left( \frac{x}{\theta} \right)^{\beta}\]

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

dist_stacy.sf(x)

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

\[\text{sf}_X(x) = Q(\alpha, z) \quad \text{for } \beta/\theta < 0 \quad \text{where } z = \left( \frac{x}{\theta} \right)^{\beta}\]

\[\text{sf}_X(x) = P(\alpha, z) \quad \text{for } \beta/\theta > 0, \quad \text{where } z = \left( \frac{x}{\theta} \right)^{\beta}\]

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

dist_stacy.qtf(q)

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

\[\text{qtf}_X(q) = \theta \cdot z^{1/\beta}; z = P^{-1}(\alpha, q) \quad \text{for } \beta/\theta > 0\]

\[\text{qtf}_X(q) = \theta \cdot z^{1/\beta}; z = Q^{-1}(\alpha, q) \quad \text{for } \beta/\theta < 0\]

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

dist_stacy.isf(q)

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

\[\text{isf}_X(q) = \theta \cdot z^{1/\beta}; z = Q^{-1}(\alpha, q) \quad \text{for } \beta/\theta > 0\]

\[\text{isf}_X(q) = \theta \cdot z^{1/\beta}; z = P^{-1}(\alpha, q) \quad \text{for } \beta/\theta < 0\]

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

dist_stacy.c_x(t)

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

\[C_X(t) = ??\]

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

dist_stacy.m_x(t)

Returns NaN, since the moment generating function does not exist.

dist_stacy.k_x(t, k=0)

Returns NaN, since the cumulant generating function does not exist.

dist_stacy.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Stacy distribution. The rth moments only exists for \(n_2 > 2r\).

\[\mu'_X(r) = ??\]

Standard moments:

\[\mu'_X(r) = \frac{\Gamma(\alpha + r/\beta)}{\Gamma(\alpha)}, \quad \alpha + r/\beta) > 0\]

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

dist_stacy.cumulants(k)

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

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