Distribution of the logarithm of a \(\chi^2\) random variable

class ctx.dist_logrv_chisquared(n)

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

The distribution of the logarithm of a \(\chi^2\) random variable with \(n\) degrees of freedom is a continuous probability distribution with the support interval \((-\infty,+\infty)\). See also Witkovský [1615].

dist_logrv_chisquared.pdf(x)

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

\[\text{pdf}_X(x) = e^x f_{\chi^2}\left(e^x, n\right),\]

Here \(f_{\chi^2}(\cdot,n)\) denotes the pdf of a random variable following an chi-squared distribution with \(n\) degress of freedom.

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

dist_logrv_chisquared.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable:

\[\text{cdf}_X(x) = F_{\chi^2}\left(e^x, n\right) = P(n/2, e^{x/2}),\]

Here \(F_{\chi^2}(\cdot,n)\) denotes the cdf of a random variable following an chi-squared distribution with \(n\) degress of freedom, and \(P(\cdot)\) is the lower regularized gamma function.

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

dist_logrv_chisquared.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable:

\[\text{sf}_X(x) = 1 - F_{\chi^2}\left(e^x, n\right) = Q(n/2, e^{x/2}).\]

Here \(1-F_{\chi^2}(\cdot,n)\) denotes the sf of a random variable following an chi-squared distribution with \(n\) degress of freedom, and \(Q(\cdot)\) is the upper regularized gamma function.

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

dist_logrv_chisquared.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable:

\[\text{qtf}_X(q) = \log\left(2 P^{-1}(n/2, q)\right).\]

Here \(P^{-1}(\cdot)\) denotes the inverse of the lower regularized incomplete gamma function.

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

dist_logrv_chisquared.isf(q)

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

\[\text{qtf}_X(q) = \log\left(2 Q^{-1}(n/2, q)\right).\]

Here \(Q^{-1}(\cdot)\) denotes the inverse of the upper regularized incomplete gamma function.

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

dist_logrv_chisquared.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable:

\[C_X(t) = 2^{it} \frac{\Gamma(n/2 +it)}{\Gamma(n/2)}\]

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

dist_logrv_chisquared.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable:

\[M_X(t) = 2^{t} \frac{\Gamma(n/2 +t)}{\Gamma(n/2)}\]

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

dist_logrv_chisquared.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 logarithm of a \(\chi^2\) random variable:

\[K_X(t) = t \cdot \log(2) + \log\left(\Gamma(n/2+t)\right) - \log\left(\Gamma(n/2)\right)\]

\[K_X^{(1)}(t) = \log(2) + \psi^{(0)}(n/2 + t),\]

\[K_X^{(j)}(t) = \psi^{(j-1)}(n/2 + t), \quad j \ge 2.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", logrv_chisquared(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_logrv_chisquared.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable: the moments are calculated from the cumulants.

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

dist_logrv_chisquared.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of the logarithm of a \(\chi^2\) random variable:

\[\kappa_{1} = \log(2) + \psi^{(0)}(n/2),\]

\[\kappa_{r} = \psi^{(r-1)}(n/2), \quad r \ge 2,\]

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

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

Approximations

ctx.logrv_chisquared_ecf(x, f, results='cdf')

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

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

ctx.logrv_chisquared_ecf_inv(q, f, results='qtf')

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

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