Boost: Noncentral \(\chi^2\) distribution

The following functions return the pdf, cdf, qtf or boost class of the noncentral chi-squared distribution with degrees of freedom \(n>0\), noncentrality parameter \(\lambda_1\), and support interval \((0, +\infty)\).

See also Wikipedia [1256], MathWorld [881], Patnaik [487], Penev and Raykov [489], Wang and Gray [866], Winterbottom [1605], BoostMath [70], Johansson [406].

Ctx.chi_squared_nc_pdf(x, n, lambda1)

where Ctx is Math53 or CtxBoost.

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

(1)\[\text{pdf}(x) = f_{\chi^2}\left(n, x; \lambda_1\right) =\frac {1}{2}e^{-(x+\lambda )/2} \left(\frac {x}{\lambda } \right)^{k/4-1/2} I_{k/2-1}({\sqrt {\lambda x}})\]

where \(I_{k}(y)\) is a modified Bessel function of the first kind of order \(k\).

Alternatively, the pdf can be written in a form which shows the relationship to the central distribution more clearly:

(2)\[\text{pdf}(x) = f_{\chi^2}\left(n, x; \lambda_1\right) = f_{\chi^2}(x, n) \times e^{-\lambda_1/2} \times {}_0F_1 \left(-; \frac{n}{2}; \frac{x \lambda_1}{4}\right).\]

Here \(f_{\chi^2}(\cdot)\) is the PDF of the central chi-square distribution, and \({}_0F_1(\cdot)\) is the confluent hypergeometric limit function.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("Chi2NcPdf(x, a, b): ", Chi2NcPdf(x, a, b))
>>> print ("dist_chi_squared_nc(a, b).pdf(x): ", dist_chi_squared_nc(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.chi_squared_nc_cdf(x, n, lambda1)

where Ctx is Math53 or CtxBoost.

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

\[\text{cdf}(x) = F_{\chi^2}\left(n, x; \lambda\right) = \int_{0}^{x} f_{\chi^2}\left(n, x; \lambda_1\right) \mathrm{d}t.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("Chi2NcCdf(x, a, b): ", Chi2NcCdf(x, a, b))
>>> print ("dist_chi_squared_nc(a, b).cdf(x): ", dist_chi_squared_nc(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.chi_squared_nc_qtf(q, n, lambda1)

where Ctx is Math53 or CtxBoost.

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

There is no known closed exact form for \(\text{qtf}(q)\). The default method is to call the function provided by Boost.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("Chi2NcQtf(q, a, b): ", Chi2NcQtf(q, a, b))
>>> print ("dist_chi_squared_nc(a, b).qtf(q): ", dist_chi_squared_nc(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_chi_squared_nc(n, lambda1)

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

The noncentral chi-square distribution is a continuous probability distribution with degrees of freedom \(n>0\), noncentrality parameter \(\lambda_1\), and support interval \((0, \infty)\). See also Wikipedia [1256], MathWorld [881], Patnaik [487], Penev and Raykov [489], Wang and Gray [866], Winterbottom [1605], BoostMath [70], Witkovský [1625], Johansson [406], R (Statistical System) [542], Yu [1653].

See also: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.chndtr.html#scipy.special.chndtr

See also: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.chndtridf.html#scipy.special.chndtridf

See also: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.chndtrinc.html#scipy.special.chndtrinc

See also: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.chndtrix.html#scipy.special.chndtrix

dist_chi_squared_nc.pdf(x)

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

\[\text{pdf}_X(x) = f_{\chi^2}\left(n, x; \lambda\right) = e^{-\lambda/2} f_{\chi^2}(n, x) {}_0F_1 \left(-; \frac{n}{2}; \frac{x \lambda}{4}\right),\]

Here \(f_{\chi^2}(\cdot)\) and \(F_{\chi^2}(\cdot)\) are the PDF and CDF, respectively, of the central chi-square distribution, and \({}_0F_1(\cdot)\) is the confluent hypergeometric limit function.

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

dist_chi_squared_nc.cdf(x)

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

\[\text{cdf}_X(x) = \int_{0}^{x} \text{pdf}_X(x) \mathrm{d} t = F_{\chi^2}\left(n, x; \lambda\right) = e^{-\lambda/2} \sum_{j=0}^\infty {\frac{(\lambda /2)^j}{j!} F_{\chi^2}\left(n+2+j, x\right) },\]

Here \(f_{\chi^2}(\cdot)\) and \(F_{\chi^2}(\cdot)\) are the PDF and CDF, respectively, of the central chi-square distribution, and \({}_0F_1(\cdot)\) is the confluent hypergeometric limit function.

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

dist_chi_squared_nc.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

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

dist_chi_squared_nc.qtf(q)

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

There is no known closed form for the quantile function \(\text{cdf}^{-1}_X(q)\): It is computed with Newton iterations where the starting values are from a central chi-square approximation.

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

dist_chi_squared_nc.isf(q)

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

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

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

dist_chi_squared_nc.c_x(t)

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

\[C_X(t) = \exp \left(\frac{i \lambda t}{1-2it}\right) (1-2it)^{-n/2}.\]

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

dist_chi_squared_nc.m_x(t)

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

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

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

dist_chi_squared_nc.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 non-central chi-squared distribution:

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

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

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

dist_chi_squared_nc.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a non-central 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_nc(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_chi_squared_nc.cumulants(k)

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

\[\kappa_{r} = 2^{r-1} (r-1)! (n+r\lambda)\]

def MakeNoncentralChiSquaredCumulants(self):
    df = iv.mpf(2000)
    lambda_ = 33
    k = 10
    kappa = iv.matrix(k+1, 1)
    kappa[0] = 1
    kappa[1] = df + lambda_
    for i in range(2, k+1):
        kappa[i] = kappa[i - 1] * 2 * (i - 1) * (1 + lambda_ / (df + (i - 1) * lambda_))
    return kappa

Recurrences: Non-central Chi-square

ctx.chi_squared_nc_recurrence(x, n, lambda)

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

The following recurrence relations hold for the pdf and CDF (see Cohen [190]):

\[f_{\chi^2}\left(n+4,x;\lambda\right) = \frac{x \cdot f_{\chi^2}\left(n,x;\lambda\right) - n \cdot f_{\chi^2}\left(n+2,x;\lambda\right) }{\lambda}\]

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

\[F_{\chi^2}\left(n,x;\lambda\right) - F_{\chi^2}\left(n-2,x;\lambda\right) = 2 \frac{\partial}{\partial \lambda} F_{\chi^2}\left(n-2,x;\lambda\right)\]

Ref:

Pav 2015: Moments of log noncentral chisquare

Yu 2011: mode of noncentral chi-square

Approximations

ctx.chi_squared_nc_gp(x, n, lambda, 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_nc_ecf(x, f, lambda1, results='cdf')

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

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

ctx.chi_squared_nc_ecf_inv(q, f, lambda1, results='qtf')

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

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

ctx.chi_squared_nc_spa(x, n, lambda, results='cdf')

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

Calculates the Luggannini-Rice saddlepoint approximation of the pdf.

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:

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

def NonCentralChi2_SPA2(self, n0, x0, lambda0_):
    n = iv.mpf(n0)
    x = iv.mpf(x0)
    lambda_ = iv.mpf(lambda0_)
    s = -(1 / (4 * x)) * (n - 2 * x + iv.sqrt(n * n + 4 * x * lambda_))
    Order = 18
    kderiv = iv.matrix(Order+2, 1)
    for j in range(0, Order+1):
        kderiv[j] = self.NonCentralChi2_CGF_Derivative(s, n, lambda_, j)

    LeftTail, RightTail = self.LugannaniRice(Order, kderiv, s)
    return LeftTail, RightTail

The following code tests the procedure

def LugannaniRiceDemo(self):
    nu = 40.0
    x = 61.0
    nc = 70.0
    LeftTail, RightTail = self.NonCentralChi2_SPA2(nu, x, nc)
    print("LeftTail:  ", LeftTail)
    print("RightTail: ", RightTail)

ctx.chi_squared_nc_spa_inv(x, n, lambda, results='qtf')

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

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