61 lines
2.4 KiB
Go
61 lines
2.4 KiB
Go
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// Copyright (c) 2026 Petr Balvín <opensource@petrbalvin.org> (https://petrbalvin.org)
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// SPDX-License-Identifier: MIT
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// Pin for the negative binomial's far regime: when the summation's seed
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// p^r underflows to an exact zero the recurrence can no longer start,
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// and the CDF answers through the regularised beta identity instead.
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// The law with r = 100 and p = 1e-4 has its mean at 999 900, so the CDF
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// at the mean is near one half; the seed 1e-400 underflows outright, and
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// the summation that ignored that answered an exact zero at every k.
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package stats
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import (
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"math"
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"testing"
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)
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func TestNegativeBinomialFarTail(t *testing.T) {
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atMean, err := NegativeBinomialCDF(999900, 100, 1e-4)
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if err != nil {
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t.Fatalf("NegativeBinomialCDF(999900, 100, 1e-4): %v", err)
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}
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// The mean is r(1−p)/p = 999 900 and the spread is near 100 000, so
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// the true value at the mean sits within a few 1e-3 of one half
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// (exactly 0.5133 by the exact referent). The identity's accuracy in
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// this regime is bounded by the lgamma noise of its front factor, so
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// the pin holds a loose band and a strict ordering.
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if math.Abs(atMean-0.5133007914) > 1e-6 {
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t.Fatalf("NegativeBinomialCDF at the mean = %.17g, want 0.5133007914 to within 1e-6", atMean)
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}
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below, err := NegativeBinomialCDF(950000, 100, 1e-4)
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if err != nil {
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t.Fatalf("NegativeBinomialCDF(950000, 100, 1e-4): %v", err)
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}
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above, err := NegativeBinomialCDF(1050000, 100, 1e-4)
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if err != nil {
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t.Fatalf("NegativeBinomialCDF(1050000, 100, 1e-4): %v", err)
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}
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if !(below < atMean && atMean < above) {
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t.Fatalf("the CDF is not increasing through the mean: %.17g, %.17g, %.17g", below, atMean, above)
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}
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far, err := NegativeBinomialCDF(2000000, 100, 1e-4)
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if err != nil {
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t.Fatalf("NegativeBinomialCDF(2000000, 100, 1e-4): %v", err)
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}
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if far < 0.9999 {
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t.Fatalf("NegativeBinomialCDF two hundred standard deviations out = %.17g, want ~1", far)
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}
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// The quantile route runs on the same CDF: the median must sit near
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// the mean the law's own moments give, a fraction of the spread
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// below it on the skewed side. The exact referent puts the median at
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// 996 569, and the pin holds it within a thousandth of the spread.
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median, err := NegativeBinomialQuantile(0.5, 1e-4, 100)
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if err != nil {
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t.Fatalf("NegativeBinomialQuantile: %v", err)
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}
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if math.Abs(median-996569) > 100 {
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t.Fatalf("NegativeBinomialQuantile(0.5) = %v, want within 100 of the exact median 996569", median)
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}
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}
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