Distribution of the negative logarithm of a beta variable

class ctx.dist_logrv_beta(a, b)

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

The distribution of the negative logarithm of a beta variable with parameters \(a>0\) and \(b>0\) is a continuous probability distribution with the support interval \((0, +\infty)\). See also Wikipedia [1238], MathWorld [868], BoostMath [57], Witkovský [1614], R (Statistical System) [545].

dist_logrv_beta.pdf(x)

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

\[\text{pdf}_X(x) = e^{-x} f_{\text{Beta}}(e^{-x}; a,b),\]

where \(f_{\text{Beta}}(\cdot, a,b)\) is the pdf of the Beta distribution with parameters \(a\) and \(b\).

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

dist_logrv_beta.cdf(x)

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

\[\text{cdf}_X(x) = 1- F_{\text{Beta}}(e^{-x}; a,b),\]

where \(F_{\text{Beta}}(\cdot, a,b)\) is the cdf of the Beta distribution with parameters \(a\) and \(b\).

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

dist_logrv_beta.sf(x)

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

\[\text{sf}_X(x) = F_{\text{Beta}}(e^{-x}; a,b),\]

where \(F_{\text{Beta}}(\cdot, a,b)\) is the cdf of the Beta distribution with parameters \(a\) and \(b\).

>>> 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_logrv_beta.qtf(q)

Returns \(\text{qtf}_X(q)\), the quantile function (qtf) of a random variable \(X\), following the distribution of the negative logarithm of a beta variable:

\[\text{qtf}_X(q) = -\log \left( F^{-1}_{\text{Beta}}(1-q; a,b) \right),\]

where \(F^{-1}_{\text{Beta}}(\cdot, a,b)\) is the qtf of the Beta distribution with parameters \(a\) and \(b\).

>>> 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_logrv_beta.isf(q)

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

\[\text{isf}_X(q) = -\log \left( F^{-1}_{\text{Beta}}(q; a,b) \right),\]

where \(F^{-1}_{\text{Beta}}(\cdot, a,b)\) is the qtf of the Beta distribution with parameters \(a\) and \(b\).

>>> 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_logrv_beta.c_x(t)

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

\[C_X(t) = \frac{\Gamma\left((a-it)\right) \Gamma\left((a+b)\right)}{\Gamma\left(a\right) \Gamma\left(a+b-it\right)}\]

>>> 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_logrv_beta.m_x(t)

Returns the moment generating function of a random variable \(X\), following the distribution of the negative logarithm of a beta variable.

\[M_X(t) = \frac{\Gamma\left((a-t)\right) \Gamma\left((a+b)\right)}{\Gamma\left(a\right) \Gamma\left(a+b-t\right)}\]

>>> 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_logrv_beta.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function, and its \(j^{\text{th}}\) derivatives, \(K_X^{(j)}(t), j = 1 \ldots k\), of a random variable \(X\), following the distribution of the negative logarithm of a beta variable.

\[K_X(t) = \log \left(\Gamma(a-t)\right) - \log \left(\Gamma(a+b-t)\right) +\log\left(\Gamma(a+b)\right) -\log\left(\Gamma(a)\right).\]

\[K^{(j)}_X(t) = (-1)^j \left( \psi^{(j-1)}(a-t) - \psi^{(j-1)}(a+b-t) \right)\]

where \(\psi^{(j)}(\cdot)\) is the polygamma function of order \(j\).

>>> 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_logrv_beta.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of the negative logarithm of a beta variable. The moments are calculated from the cumulants.

>>> 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_logrv_beta.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of the negative logarithm of a beta variable.

\[\kappa_j = (-1)^j \left( \psi^{(j-1)}(a) - \psi^{(j-1)}(a+b) \right)\]

where \(\psi^{(j)}(\cdot)\) is the polygamma function of order \(j\).

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

Approximations

ctx.logrv_beta_ecf_pdf(x, a, b, results='cdf')

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

Calculates the Edgeworth approximation to the pdf, cdf and sf.

ctx.logrv_beta_ecf_qtf(q, a, b, results='cdf')

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

Calculates the Cornish-Fisher approximation to the qtf and isf.