scipy.special.ncfdtr#

scipy.special.ncfdtr(dfn, dfd, nc, f, out=None) = <ufunc 'ncfdtr'>#

Cumulative distribution function of the non-central F distribution.

The non-central F describes the distribution of,

\[Z = \frac{X/d_n}{Y/d_d}\]

where \(X\) and \(Y\) are independently distributed, with \(X\) distributed non-central \(\chi^2\) with noncentrality parameter nc and \(d_n\) degrees of freedom, and \(Y\) distributed \(\chi^2\) with \(d_d\) degrees of freedom.

Parameters:

dfnarray_like: Degrees of freedom of the numerator sum of squares. Range (0, inf).
dfdarray_like: Degrees of freedom of the denominator sum of squares. Range (0, inf).
ncarray_like: Noncentrality parameter. Range [0, inf).
farray_like: Quantiles, i.e. the upper limit of integration.
outndarray, optional: Optional output array for the function results

Returns:

cdfscalar or ndarray: The calculated CDF. If all inputs are scalar, the return will be a float. Otherwise it will be an array.

See also

ncfdtri: Quantile function; inverse of ncfdtr with respect to f.
ncfdtridfd: Inverse of ncfdtr with respect to dfd.
ncfdtridfn: Inverse of ncfdtr with respect to dfn.
ncfdtrinc: Inverse of ncfdtr with respect to nc.
scipy.stats.ncf: Non-central F distribution.

Notes

This function calculates the CDF of the non-central f distribution using the Boost Math C++ library [1].

The cumulative distribution function is computed using Formula 26.6.20 of [2]:

\[F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2} \frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}),\]

where \(I\) is the regularized incomplete beta function, and \(x = f d_n/(f d_n + d_d)\).

Note that argument order of ncfdtr is different from that of the similar cdf method of scipy.stats.ncf: f is the last parameter of ncfdtr but the first parameter of scipy.stats.ncf.cdf.

References

[1]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

[2]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples

>>> import numpy as np
>>> from scipy import special
>>> from scipy import stats
>>> import matplotlib.pyplot as plt

Plot the CDF of the non-central F distribution, for nc=0. Compare with the F-distribution from scipy.stats:

>>> x = np.linspace(-1, 8, num=500)
>>> dfn = 3
>>> dfd = 2
>>> ncf_stats = stats.f.cdf(x, dfn, dfd)
>>> ncf_special = special.ncfdtr(dfn, dfd, 0, x)

>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, ncf_stats, 'b-', lw=3)
>>> ax.plot(x, ncf_special, 'r-')
>>> plt.show()

../../_images/scipy-special-ncfdtr-1.png