Home › Stats › Overview

Distribution tails — Lodestar.Stats

One type, Distributions: the density, both tails and both inverses of the standard normal, Student's t, F and chi-squared laws, for a caller holding its own statistic, without going through a hypothesis test to reach them.

Why five members first, and twenty now

This package computes every tail it needs from its own log-gamma, incomplete beta and incomplete gamma, and decisions/0003 kept that layer internal — deliberately, and with the condition for changing its mind written down: "publishing later stays possible the day a second package needs the same functions."

Lodestar.Stats.Regression is that day. What it needs is a Student tail for each coefficient's p-value, a Student quantile for each confidence interval, and an F tail for the overall test. Those three are published. The log-gamma, the incomplete beta and gamma, the normal tail and the finite-sample Kolmogorov distribution underneath them stay internal, because nothing has asked for them and an unpublished API can still be published later — the reverse is not true.

The other fifteen

A statistics package at 1.0 that computes these laws and hands out five of their functions leaves a caller writing their own test, interval or power calculation to find t.cdf or norm.ppf elsewhere. #1158 published the rest: Pdf, Cdf, Sf, Quantile and Isf for each of the four laws. It needed one new piece of numerics, the inverse of the incomplete gamma behind the chi-squared quantiles, and it fixed one defect of the members already out: Distributions.StudentQuantile stopped at 1.3e154, where squaring the statistic overflowed, so the Cauchy's 1e-300 point, 3.18e299, was out of reach. The two quantiles published first also answer 0 and 1 now, as scipy's do.

What publishing cost

A parity promise of its own. The internal tests check these functions against closed forms, which is enough for a p-value near 0.05 and says nothing about the far tail. Two corpora replay scipy.stats, compared relatively, since an absolute 1e-9 would accept an implementation returning zero: the first down to 3.1e-24, and the second, for the twenty members, over a grid reaching 1e-300 on either side. That grid keeps a scipy quantile only where scipy's own tail, read at it, gives the probability back: at p = 1e-8, f.isf is 5e-9 off its closed form, and several inverses clamp in the far tail to a value whose tail is another probability entirely.

It also cost a correction. The internal helper behind Distributions.StudentQuantile solves P(T > x) = p and therefore carries the opposite sign to a quantile. Published under its own name unchanged, it would have handed a caller -2.18 where every table prints +2.18.

Types

Type	What it is
Distributions	Pdf, Cdf, Sf, Quantile and Isf of the normal, t, F and chi-squared laws.

See also

• Hypothesis tests — the ten families built on the same machinery.
• Python → C# equivalence.