Generalized Beta (Type 1) distribution

class ctx.dist_genbeta1(a, b, p, q)

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

The Generalized Beta (Type 1) 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 230), Wikipedia [1300].

dist_genbeta1.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a Generalized Beta (Type 1) distribution:

\[\text{pdf}_X(x) = \frac{a x^{a p -1} [1- (x/b)^a]^{q-1} }{b^{ap} B(p,q) }, \quad 0 \le x \le b.\]

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

dist_genbeta1.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Generalized Beta (Type 1) distribution:

\[\text{cdf}_X(x) = I_z(p,q), \quad \text{where } z = \left( \frac{y}{b} \right)^a, y = \left( \frac{x^a}{1+x^a} \right)^{1/a} \quad x>0, \quad 0 \le x \le b.\]

See also: Kleiber, page 231

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

dist_genbeta1.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a Generalized Beta (Type 1) distribution:

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

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 = 1; x = 3;
>>> print (" sf: ", fisher_f(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_genbeta1.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Generalized Beta (Type 1) 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.

MISSING: transformatio from x to y.

>>> 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_genbeta1.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a Generalized Beta (Type 1) distribution:

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

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

MISSING: transformatio from x to y.

>>> 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_genbeta1.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Generalized Beta (Type 1) 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_genbeta1.m_x(t)

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

dist_genbeta1.k_x(t, k=0)

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

dist_genbeta1.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Generalized Beta (Type 1) distribution. The rth moments only exists for \(-ap < k < \infty\).

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

See Kleiber, page 231.

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

dist_genbeta1.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Generalized Beta (Type 1) 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