Boost: Noncentral Fisher F distribution

Boost: Noncentral Fisher \(F\) distribution#

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

See also Wikipedia [1254], MathWorld [882], BoostMath [71], Benton and Krishnamoorthy [35], Butler and Paolella [173], Chou [186], Chattamvelli and Jones [182], Wang and Gray [866], Johansson [406].

Ctx.fisher_f_nc_pdf(x, m, n, lambda1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the non-central Fisher F distribution:

(1)#\[\text{pdf}(x) = f_{\text{FisherF}}(x;m, n;\lambda_1) = \frac{m}{n} \int_{0}^{\infty} y \cdot f_{\chi^2} \left(x \cdot y \cdot m/n, m, \lambda_1\right) \cdot f_{\chi^2}(y, n) \: \mathrm{d}y\]

Here \(f_{\chi^2}(\cdot, m, \lambda_1)\) denotes the pdf of the noncentral \(\chi^2\) distribution with \(m\) degrees of freedom and noncentrality parameter \(\lambda_1\) (see chi2_nc_pdf()), and \(f_{\chi^2}(\cdot, n)\) denotes the pdf of the central \(\chi^2\) distribution with \(n\) degrees of freedom (see chi_squared_pdf()).

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

(2)#\[\text{pdf}(x) = f_{F'}(x;n_1,n_2,\lambda_1) = e^{-\lambda_1/2} f_{F}(x;n_1,n_2) {}_1F_1 \left(\tfrac{1}{2}(m+n), \tfrac{1}{2}n, \tfrac{n x \lambda_1}{2(m+n x)}\right),\]

Here \(f_{F}(\cdot)\) denotes the pdf of the central \(F\)-distribution, and \({}_1F_1(\cdot)\) is the confluent hypergeometric function.

The following example shows both forms of the syntax:

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

Ctx.fisher_f_nc_cdf(x, m, n, lambda1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the non-central Fisher F distribution:

(3)#\[\text{cdf}(x) = F_{\text{FisherF}}(x;m, n;\lambda_1) = \frac{m x}{n}\int_{0}^{\infty} f_{\chi^2} \left(x \cdot y \cdot m/n, m; \lambda_1\right) \cdot [1-F_{\chi^2}(y, n)] \: \mathrm{d}y\]

Here \(f_{\chi^2}(\cdot, m, \lambda_1)\) denotes the pdf of the noncentral \(\chi^2\) distribution with \(m\) degrees of freedom and noncentrality parameter \(\lambda_1\) (see chi2_nc_pdf()), and \(F_{\chi^2}(\cdot, n)\) denotes the cdf of the central \(\chi^2\) distribution with \(n\) degrees of freedom (see chi_squared_cdf()).

The following example shows both forms of the syntax:

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

Ctx.fisher_f_nc_qtf(q, m, n, lambda1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the non-central Fisher F distribution:

There is no known closed exact form for \(\text{qtf}(q)\). For fpm. 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 ("FisherFNcQtf(q, a, b): ", FisherFNcQtf(q, a, b))
>>> print ("dist_fisher_f_nc(a, b).qtf(q): ", dist_fisher_f_nc(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_fisher_f_nc(m, n, lambda1)#

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

The non-central Fisher F distribution is a continuous probability distribution with \(m>0\) and \(n>0\) degrees of freedom, noncentrality parameter \(\lambda_1\) , and the support interval \((0, +\infty)\). See also Wikipedia [1254], MathWorld [882], BoostMath [71], Benton and Krishnamoorthy [35], Butler and Paolella [173], Chou [186], Chattamvelli and Jones [182], Wang and Gray [866], Witkovský [1626], Johansson [406], R (Statistical System) [543]. Also doubly-noncentral: Yin [1652].

dist_fisher_f_nc.pdf(x)#

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

\[\text{pdf}_X(x) = f_{F'}(x;n_1,n_2,\lambda) = e^{-\lambda/2} f_{F}(x;n_1,n_2) {}_1F_1 \left(\tfrac{1}{2}(m+n), \tfrac{1}{2}n, \tfrac{n x \lambda}{2(m+n x)}\right),\]

Here \(f_{F}(\cdot)\) denotes the pdf of the central \(F\)-distribution, and \({}_1F_1(\cdot)\) is the confluent hypergeometric function.

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

dist_fisher_f_nc.cdf(x)#

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

\[\text{cdf}_X(x) = F_{F'}(x;m,n,\lambda) = e^{-\lambda} \sum_{j=0}^{\infty}{\frac{(\lambda/2)^j}{j!}F_F(x; m+2j,n)}\]

Here \(F_{F}(\cdot)\) denotes the cdf of the central \(F\)-distribution.

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

dist_fisher_f_nc.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a non-central Fisher F 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: ", fisher_f_nc(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_fisher_f_nc.qtf(q)#

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

There is no known explicit form for the quantile function \(\text{cdf}^{-1}_X(x)\): It is computed using Newton iterations with starting values from a central \(F\) approximation.

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

dist_fisher_f_nc.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a non-central Fisher F 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: ", fisher_f_nc(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_fisher_f_nc.c_x(t)#

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

\[C_X(t) = \int_{0}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]

\[\phi(t) = e^{-\lambda/2} \sum_{k=0}^{\infty} \frac{(\lambda/2)^k}{k!} {}_1F_1 \left(\frac{\nu_1}{2}+k, -\frac{\nu_2}{2}, -\frac{\nu_2}{\nu_1} it \right).\]

CRC: page 122: arcsin, page 146 half-normal

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

dist_fisher_f_nc.m_x(t)#: Returns NaN, since the moment generating function does not exist.

dist_fisher_f_nc.k_x(t, k=0)#: Returns NaN, since the cumulant generating function does not exist.

dist_fisher_f_nc.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a non-central Fisher F distribution. The rth moment only exists for \(n_2 > 2r\) and is given by

\[\mu'_X(r) = \left(\frac{n_2}{n_1}\right)^{r} \frac{\Gamma( \tfrac{1}{2}n_1+r) \Gamma( \tfrac{1}{2}n_2-r)}{\Gamma( \tfrac{1}{2}n_2)} {}_1\widetilde{F}_1(-r; \tfrac{1}{2}n_1; -\tfrac{1}{2}\lambda_1),\]

where \({}_1\widetilde{F}_1(a,b;z)\) denotes Kummer’s regularized confluent hypergeometric function.

See also: Paoella 2, page 358-360

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

dist_fisher_f_nc.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a non-central Fisher F distribution. The cumulants are calculated from the moments.

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

Additional methods: Confidence intervals and sample size estimates

dist_fisher_f_nc.nc_ci(alpha, beta)#

Returns a confidence interval for the noncentrality parameter lambda

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

Recurrences: Non-central Fisher F, recurrence pdf

ctx.fisher_f_nc_pdf_recurrence(x, lambda, start_n1, start_n2, target_n1, target_n2)#

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

The following recurrence relations hold for the pdf:

Let the density \(g_{m,n}\) be that of \(m/n\) times an \(F_{m,n}\) random variable. Let \(G_{m,n}(y)\) be its distribution function, and let \(g_{m,n}^{\lambda}\) and \(G_{m,n}^{\lambda}(y)\) be the density and distribution function of its (singly) noncentral version (the distribution of \(\chi_m^2(\lambda)/\chi_n^2(0)\)). Then the following recurrence relations hold (see Chattamvelli and Jones [182])

Applies a recurrence relation to calculate the cdf for different degrees of freedom for a given value of x. This is mostly useful when using asymptotic methods.

\[n\left[G_{m,n+2}^{\lambda}(y)-G_{m-2,n+2}^{\lambda}(y)\right] = -2g_{m,n}^{\lambda}(y)\]

\[\lambda(1+y) g_{m+4,n}^{\lambda}(y) = [\lambda y - m(1+y)]g_{m+2,n}^{\lambda}(y) + y(m+n)g_{m,n}^{\lambda}(y).\]

\[n(1+y) g_{m,n+2}^{\lambda}(y) = (m+n)g_{m,n}^{\lambda}(y) + \lambda g_{m+2,n}^{\lambda}(y).\]

\[\lambda g_{m+4,n-2}^{\lambda}(y) + m g_{m+2,n-2}^{\lambda}(y) = (n-2) y g_{m,n}^{\lambda}(y).\]

Recurrences: Non-central Fisher F, recurrence cdf

ctx.fisher_f_nc_cdf_recurrence(x, lambda, start_n1, start_n2, target_n1, target_n2)#

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

Applies a recurrence relation to calculate the cdf for different degrees of freedom for a given value of x. This is mostly useful when using asymptotic methods.

From these equations we obtain

\begin{eqnarray} \lambda(1+y) G_{m+6,n}^{\lambda}(y) & = & [\lambda y - (m+2-\lambda )(1+y)]G_{m+4,n}^{\lambda}(y) \\ & + & [(m+2)(1+y)+y(m+n-\lambda)]G_{m+2,n}^{\lambda}(y) \nonumber \\ & - & y(m+n)G_{m,n}^{\lambda}(y) \nonumber \end{eqnarray}

\begin{eqnarray} n(1+y) :cite:t:`G_{m,n+2}^{\lambda}(y)-G_{m+2,n+2}^{\lambda}(y)] & = & (m+n)G_{m,n}^{\lambda}(y) \\ & + & (\lambda-m-n)G_{m+2,n}^{\lambda}(y) \nonumber \\ & - & \lambda G_{m+4,n}^{\lambda}(y) \nonumber \end{eqnarray}

\begin{eqnarray} (n-2)y :cite:t:`G_{m,n}^{\lambda}(y)-G_{m+2,n}^{\lambda}(y)] & = & (m+2)G_{m+2,n-2}^{\lambda}(y) \\ & + & (\lambda-m-2)G_{m+4,n-2}^{\lambda}(y) \nonumber \\ & - & \lambda G_{m+6,n-2}^{\lambda}(y) \nonumber \end{eqnarray}

Approximations

ctx.fisher_f_nc_ecf(x, m, n, lambda1, results='cdf')#

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

Calculates the Edgeworth approximation to the pdf, cdf and sf. See also: MathWorld [882], Paolella [479], page 358-360.

ctx.fisher_f_nc_ecf_inv(q, m, n, lambda1 results='qtf')#

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

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

Boost: Noncentral Fisher F distribution

Contents

Boost: Noncentral Fisher \(F\) distribution#