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Distributions.NormalQuantile

The value a standard normal falls below with probability p.

public static double NormalQuantile(double p)

Parameters — p is a probability in [0, 1]. 0 answers −∞ and 1 answers +∞, as scipy's do; they were refused until #1158.

Returns — scipy.stats.norm.ppf(p).

Exceptions — ArgumentOutOfRangeException when p is outside [0, 1], NaN included.

Example — the multiplier a large-sample confidence interval takes.

using Lodestar.Stats;

double ninetyFive = Distributions.NormalQuantile(0.975);  // => 1.959963…
double ninetyNine = Distributions.NormalQuantile(0.995);  // => 2.575829…
double median = Distributions.NormalQuantile(0.5);  // => 0

Remarks — this is the quantile, not the inverse survival function: P(Z ≤ z) = p, where Distributions.NormalIsf solves P(Z > z) = p. It is scipy.special.ndtri, the function norm.ppf calls, ported statement for statement from Cephes' rational approximations, so the value is scipy's to the last bit on both tails and the centre where .NET's logarithm is glibc's, as on Linux, and to 1e-14 relative elsewhere (#1684); at the median it is exactly zero.

Why this exists rather than a Student quantile at a large degrees of freedom. Student converges on the normal, so the obvious substitute is StudentQuantile(p, df) with df very large. Measured against scipy.stats.norm.ppf(0.975), its relative error is the two laws' own distance, about 1.2 / df until rounding takes over:

df	relative error
1e6	1.2e-6
1e8	1.2e-8
1e9	1.2e-9
1e12	1.2e-12
1e16	1.1e-16

Past that rounding decides: one ulp off at 1e16, two at 1e17 and at 9.9e19, and exact from df = 1e20, where the Student law is the normal one. A Kaplan-Meier confidence bound is built on the log-log transform of the estimate, which amplifies a smaller df's distance into the digits of a bound its corpus compares at 1e-9. This member answers to the last bit on Linux instead. Decision 0003 publishes a member once another package names it, and Lodestar.Survival, Lodestar.Preprocessing, Lodestar.Stats.Regression and Lodestar.Stats.TimeSeries call this one.

Applies to — net10.0, netstandard2.0.

See also — Distributions.StudentQuantile, Distributions.ChiSquaredSf.