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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 strictly inside (0, 1). The endpoints are refused rather than answered with the two infinities.

Returns — scipy.stats.norm.ppf(p).

Exceptions — ArgumentOutOfRangeException when p is not strictly inside (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. The internal helper solves P(Z > z) = p and carries the opposite sign; the published member negates, which is the distribution's symmetry about zero rather than a correction. The same trap Distributions.StudentQuantile documents, and the same answer.

At the median it returns positive zero. Negating the helper's exact zero would otherwise hand a caller -0 from a published method — true of 0.0 == -0.0, but not something an API should print.

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), that substitute stops improving at about 1e-8 relative:

df	relative error
1e6	1.2e-6
1e7	1.2e-7
1e8	1.2e-8
1e9	1.5e-7
1e12	3.3e-5

Below the best point the convergence is incomplete; above it the Student tail being inverted loses ground. A Kaplan-Meier confidence bound is built on the log-log transform of the estimate, which amplifies that error into the seventh digit of a bound — past the 1e-9 its corpus compares at. This member answers to about 1e-15 instead. Decision 0003, as docs/guides/performance.md amended it, has the whole measurement.

Applies to — net10.0, netstandard2.0.

See also — Distributions.StudentQuantile, Distributions.ChiSquaredSf.