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Distributions.FisherSnedecorQuantile

The value an F falls below with probability p.

public static double FisherSnedecorQuantile(double p, double numeratorDf, double denominatorDf)

Parameters — p is a probability in [0, 1]; 0 answers 0 and 1 answers +∞. numeratorDf and denominatorDf are the two degrees of freedom, each of which must be positive and finite.

Returns — scipy.stats.f.ppf(p, dfn, dfd).

Exceptions — ArgumentOutOfRangeException when p is outside [0, 1], NaN included, or when either degrees of freedom is not positive.

Example — the 5% critical value of an overall test on two and twenty degrees of freedom.

using Lodestar.Stats;

double critical = Distributions.FisherSnedecorQuantile(0.95, 2.0, 20.0);   // => 3.492828…

Remarks — Inverts the incomplete beta for both y and 1 − y, refining by Newton on whichever of the two is below one half, so neither is one minus a number near one: at shapes (1e10, 1) the root sits 2e-10 below one, and forming 1 − y from it was 1e-7 off. Where scipy disagrees, scipy is checked against itself: at p = 1e-100 with dfn = 0.5 it answers 1.3e-307, whose own lower tail is 1.4e-77; the true point underflows, and this answers zero. docs/equivalence.md lists these points. It searches for the smallest double whose lower tail before its last rounding reaches p, judged on the smaller tail, one less p past one half. The contract is 8 ulps (Cyril, 2026-10-29): the tail at the answer lies within 8 ulps of its target, or the answer lies next to the crossing where one ulp of the statistic moves the tail further, past the 14 to 15 digits Algorithm 708 certifies; the smallest double reaching it is the method's aim, not its promise. The tail is the regularized incomplete beta to 106 bits, rounded once, within 8 ulps of its exact value and monotone in the statistic to within that, up to a harmonic shape ab/(a + b) of 1000, the chi-squared limit a half of the split below the normal doubles takes included. Past that shape, below shapes of 1e5 where Temme's expansion takes over, the tail is computed in doubles and steps thousands of ulps between runs of the statistic; there the answer is the inversion's root, which need not reach p. Elsewhere Newton steps on the tail before its last rounding settle the root, and a search of the doubles within 128 ulps of where they land, then of the root, then outward from the root, finds the answer. Two exceptions: where p and the root's tail are both subnormal it answers that root, the exact tail telling nothing finer than a subnormal's granule; and where the tail moves by less than an ulp of p over 2²⁰ doubles of the statistic, as near the median at dfs below about 1e-6, or the tail is a lifted df's double form, it answers the doubles next to where the Newton steps land or to the inversion's root if the crossing is there, else that root, the search costing some 40 tails (#1745). Below dfs of 6.2e-9, where the incomplete beta's first order holds at the root it gives for p, the search starts from that root, judged there on the caller's own tail rather than the smaller, 1 for the median of equal dfs and 0 where the tail at the smallest double is already past p (#1743). An inversion that overflows past the largest double answers a double where the smaller tail there reaches its target, p or one less p as that tail is the caller's or the other: the smallest double reaching it, searched as any other. A root past every double is +∞ unless at the largest double FisherSnedecorCdf rounds to p, or the smaller tail there, as it or FisherSnedecorSf rounds it, to the smaller of p and one less p: that double then (#1746).

Applies to — net10.0, netstandard2.0.

See also — Distributions.FisherSnedecorIsf, Distributions.FisherSnedecorCdf.