Generalized Beta (Type 2) distribution

Generalized Beta (Type 2) distribution#

class ctx.dist_genbeta2(a, b, p, q)#

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

The Generalized Beta (Type 2) distribution distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), \(p > 0\), \(q > 0\), and the support interval \((0, +\infty)\).

See also: Kleiber and Kotz [415] (page 184), Wikipedia [1300].

dist_genbeta2.pdf(x)#

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a Generalized Beta (Type 2) distribution:

\[\text{pdf}_X(x) = \frac{a x^{a p -1}}{b^{ap} B(p,q) [1+ (x/b)^a]^{p+q}}, \quad x>0.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; x = 3;
>>> print ("pdf: ", fisher_f(mu, sigma).pdf(x))
6.3563523462564525625625625624562356E-20

dist_genbeta2.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Generalized Beta (Type 2) distribution:

\[\text{cdf}_X(x) = I_z(p,q), \quad \text{where } z = \left( \frac{x}{b} \right)^a, \quad x>0.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; x = 3;
>>> print ("cdf: ", fisher_f(mu, sigma).pdf(x))
6.3563523462564525625625625624562356E-20

dist_genbeta2.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a Generalized Beta (Type 2) distribution:

\[\text{sf}_X(x) = 1-I_z(p,q) = I_{1-z}(p,q), \quad \text{where } z = \left( \frac{x}{b} \right)^a, \quad x>0.\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function, and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function. See Kleiber, page 188.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; x = 3;
>>> print (" sf: ", fisher_f(mu, sigma).pdf(x))
sf: 6.3563523462564525625625625624562356E-20

dist_genbeta2.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Generalized Beta (Type 2) distribution:

\[\text{qtf}_X(\text{prob}) = b \cdot z^{1/a}, \quad \text{where } z = \mathrm{ibeta\_inv}(p, q, \text{prob}).\]

Here \(\mathrm{ibeta\_inv}(\cdot)\) denotes the inverse of the real normalised incomplete beta function.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; q = 0.3;
>>> print ("qtf: ", fisher_f(mu, sigma).qtf(q))
qtf: 6.3563523462564525625625625624562356E+00

dist_genbeta2.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a Generalized Beta (Type 2) distribution:

\[\text{isf}_X(\text{prob}) = b \cdot z^{1/a}, \quad \text{where } z = \mathrm{ibetac\_inv}(p, q, \text{prob}).\]

Here \(\mathrm{ibetac\_inv}(\cdot)\) denotes the inverse of the real normalised complementary incomplete beta function.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; q = 0.3;
>>> print ("isf: ", fisher_f(mu, sigma).isf(q))
6.3563523462564525625625625624562356E+00

dist_genbeta2.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Generalized Beta (Type 2) distribution:

\[C_X(t) = ??\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; t = 0.3;
>>> print ("c_x: ", fisher_f(mu, sigma).c_x(t))
6.3563523462564525625625625624562356E+00

dist_genbeta2.m_x(t)#

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

dist_genbeta2.k_x(t, k=0)#

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

dist_genbeta2.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 2 \ldots k\), of a random variable \(X\), following a Generalized Beta (Type 2) distribution. The rth moments only exists for \(-ap < k < aq\).

\[\mu'_X(k) = \frac{b^k B(p+k/a, q-k/a)}{B(p,q)} = \frac{b^k \Gamma(p+k/a) \Gamma(q-k/a)}{\Gamma(p)\Gamma(q)},\]

See Kleiber, page 188.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).moments(k))
6.3563523462564525625625625624562356E+00

dist_genbeta2.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 2 \ldots k\), of a random variable \(X\), following a Generalized Beta (Type 2) distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).cumulants(k))
6.3563523462564525625625625624562356E+00
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