Boost: Chi-Squared distribution

Boost: Chi-Squared distribution#

The following functions return the pdf, cdf, qtf or boost class of the chi-squared distribution with \(n > 0\) degrees of freedom and the support interval \((0,+\infty)\).

See also Wikipedia [1240], MathWorld [869], BoostMath [59], Ehrhardt [309] (3.9.6)

Ctx.chi_squared_pdf(x, a=0, b=1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the chi-squared distribution:

\[\text{pdf}(x) = f_{\chi^2}\left(x, n\right) = \frac{1}{2^{n/2} \Gamma(n/2)} x^{(n-2)/2}e^{-x/2}.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ChiSquaredPdf(x, a, b): ", ChiSquaredPdf(x, a, b))
>>> print ("dist_chi_squared(a, b).pdf(x): ", dist_chi_squared(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.chi_squared_cdf(x, a=0, b=1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the chi-squared distribution:

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

Here \(P(\cdot)\) denotes the lower regularized incomplete gamma function (RealGammaP).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ChiSquaredCdf(x, a, b): ", ChiSquaredCdf(x, a, b))
>>> print ("dist_chi_squared(a, b).cdf(x): ", dist_chi_squared(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.chi_squared_qtf(q, a=0, b=1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the chi-squared distribution:

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

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

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("ChiSquaredQtf(q, a, b): ", ChiSquaredQtf(q, a, b))
>>> print ("dist_chi_squared(a, b).qtf(q): ", dist_chi_squared(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_chi_squared(n)#

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

The chi-squared distribution is a continuous probability distribution with \(n > 0\) degrees of freedom and the support interval \((0,+\infty)\). See also Wikipedia [1240], MathWorld [869], BoostMath [59], Witkovský [1615], R (Statistical System) [547].

dist_chi_squared.pdf(x)#

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

\[\text{pdf}_X(x) = f_{\chi^2}\left(n, x\right) = \frac{1}{2^{n/2} \Gamma(n/2)} x^{(n-2)/2}e^{-x/2}.\]

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

dist_chi_squared.cdf(x)#

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

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

Here \(P(\cdot)\) and \(P^{-1}(\cdot)\) are the regularized gamma function and its functional inverse, respectively.

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

dist_chi_squared.sf(x)#

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

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

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

dist_chi_squared.qtf(q)#

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

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

Here \(P(\cdot)\) and \(P^{-1}(\cdot)\) are the regularized gamma function and its functional inverse, respectively.

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

dist_chi_squared.isf(q)#

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

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

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

dist_chi_squared.c_x(t)#

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

\[C_X(t) = (1-2it)^{-n/2}.\]

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

dist_chi_squared.m_x(t)#

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

\[M_X(t) = (1-2t)^{-n/2}, \quad t \in \left(-\infty, \tfrac{1}{2}\right).\]

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

dist_chi_squared.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 a chi-squared distribution:

\[K_X(t) = - \frac{n}{2} \log(1-2t), \quad t \in \left(-\infty, \tfrac{1}{2}\right),\]

\[K_X^{(j)}(t) = \frac{2^{j-1}(j-1)!}{(1-2t)^j} n .\]

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

dist_chi_squared.moments(k)#

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

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

dist_chi_squared.cumulants(k)#

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

\[\kappa_{r+1} = 2^r r! n.\]

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

Recurrences: Central Chi-square

ctx.chi_squared_recurrence(x, n, lambda)#

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

The following recurrence relations hold for the pdf and CDF:

\[f_{\chi^2}(n+2, x) = \frac{x}{n} f_{\chi^2}(n, x)\]

\[F_{\chi^2}(n, x) - F_{\chi^2}(n+2, x) = 2f_{\chi^2}(n+2, x)\]

Approximations

ctx.chi_squared_gp(x, n, results='cdf')#

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

Calculates the pdf, cdf and sf from the characteristic function using the procedure of Gil-Pelaez (see gil_pelaez_pdf() and gil_pelaez_cdf()).

ctx.chi_squared_ecf(x, f, results='cdf')#

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

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

ctx.chi_squared_ecf_inv(q, f, results='qtf')#

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

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

ctx.chi_squared_spa(x, n, results='c')#

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

Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf.

The solution \(\hat{s}(x)\) of the saddlepoint equation \(K_X^{(1)}(\hat{s}(x))=x\), of a random variable \(X\), following a non-central chi-squared distribution is given by:

\[\hat{s}(x) = -\frac{1}{4x} \left[n-2x+n \right], \quad x>0\]

ctx.chi_squared_spa_inv(x, n, results='qtf')#

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

Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.

Boost: Chi-Squared distribution

Contents

Boost: Chi-Squared distribution#