Poisson Confidence Interval

Poisson Confidence Interval (cross-cutting/numerics/poisson-confidence-interval)

Use when you must compute an exact confidence interval for a Poisson rate from a total event count over an exposure: k defects, failures or rare events observed over T flight cycles, test hours or inspection units, giving the Garwood exact interval from the chi-square relation and the normal approximation as the large-count cross-check. All chi-square and normal quantiles come from the module's own inversion machinery, pure Python stdlib with no external statistics packages. It pairs with cross-cutting/numerics/proportion-confidence-interval for success-over-trials attribute data (no exposure term) and with cross-cutting/numerics/hypothesis-testing for the verdict layer on the same count data.

Domain quick reference

• Conventions: k events over exposure T, confidence level cl in (0, 1) (default 0.95), alpha = 1 - cl. The rate estimate is lambda_hat = k / T per exposure unit.
• Chi-square quantile: chi2(df, q) is the value x with P(chi-square with df degrees of freedom <= x) = q. The module evaluates the chi-square CDF as the regularized lower incomplete gamma P(df / 2, x / 2), computed by power series below the crossover x < a + 1 and by a Lentz continued fraction of the survival function above it, and inverts by bisection to tolerance 1e-9.
• Exact Garwood interval: lower = chi2(2k, alpha / 2) / (2T), upper = chi2(2k + 2, 1 - alpha / 2) / (2T), per exposure unit. A zero count gives lower 0.0 because the df = 0 quantile is degenerate; the upper still uses df = 2.
• Normal approximation: bounds (k +/- z * sqrt(k)) / T with z = normal_quantile((1 + cl) / 2) (1.959964 at 0.95), the Wald form that gates large-count claims.
• Small-count behavior: at k = 12 the exact upper bound exceeds the normal upper bound and the exact interval is the wider one, while the two upper bounds converge as the count grows (within 5 percent by k = 200). The Garwood interval is exact but not centered, so the lower-side comparison depends on the case; the conservative widening is on the upper bound and in total width.
• Exposure scaling: both bounds scale as 1 / T, so doubling the exposure halves the rate and tightens every bound by exactly two.
• NACA TR-824 frames the aeronautics numerics context; the relations above are standard engineering methodology, summary-only.

Workflow

1. State the count and exposure: total events k (defects, failures) over exposure T (flight cycles, test hours, inspection units), and confirm the rate estimate lambda_hat = k / T with poisson_rate(k, T).
2. Choose the confidence level cl (default 0.95) and split the two-sided tail: alpha = 1 - cl with alpha / 2 on each side for the chi-square quantiles.
3. Invert the chi-square CDF: chi_square_quantile(df, q) bisects the regularized lower incomplete gamma survival function P(df / 2, x /
   2. to tolerance 1e-9, called with df = 2k at q = alpha / 2 and with df = 2k + 2 at q = 1 - alpha / 2; a zero count skips the degenerate df = 0 lower quantile and reports lower bound 0.
4. Form the exact Garwood interval with poisson_confidence_interval(k, T, cl): lower = chi2(2k, alpha / 2) / (2T), upper = chi2(2k + 2, 1 - alpha / 2) / (2T), method "exact-poisson", and check the rate estimate lies inside the bounds.
5. Run the normal approximation cross-check with normal_approximation_interval(k, T, cl): bounds (k +/- z * sqrt(k)) / T with z = normal_quantile((1 + cl) / 2), method "normal-approximation".
6. Report the rate estimate, both intervals, the containment verdict and the convergence note: the exact upper bound exceeds the normal upper bound at small counts and the two agree within 5 percent by k = 200, so gate the defect-rate claim on the exact bounds.

Worked example

k = 12 defects over T = 240 flight cycles at cl = 0.95, module output in brackets:

• Rate estimate: lambda_hat = 12 / 240 = 0.05 per cycle (poisson_rate(12, 240) = 0.05).
• Chi-square quantiles: chi2(24, 0.025) = 12.401 and chi2(26, 0.975) = 41.923, inverted by bisection of the incomplete gamma survival function.
• Exact Garwood interval: lower = 12.401 / 480 = 0.025836, upper = 41.923 / 480 = 0.087340, method "exact-poisson".
• Normal approximation: (12 +/- 1.959964 * sqrt(12)) / 240 gives [0.021710, 0.078290], method "normal-approximation".
• Zero-count edge: k = 0, T = 100 gives exact [0.0, 0.036889] (upper from chi2(2, 0.975) / 200 = 7.378 / 200).
• Convergence: k = 200, T = 4000 gives exact upper 0.057430 against normal upper 0.056930, about 0.9 percent apart, inside the 5 percent bound.

Verification

• Confirm poisson_rate(12, 240) returns 0.05 exactly and that the exact interval rate key is the same estimate.
• Confirm poisson_confidence_interval(12, 240) returns lower 0.025836 and upper 0.087340, within 1e-4 of the spec anchors 0.02584 and 0.08734.
• Confirm normal_approximation_interval(12, 240) returns lower 0.021710 and upper 0.078290, within 2e-3 of the anchors 0.0217 and 0.0783.
• Confirm poisson_confidence_interval(0, 100) returns lower 0.0 and upper 0.036889, within 2e-4 of the anchor 0.0369, and that the zero-count upper bound is positive.
• Confirm the rate estimate lies inside the exact interval and that doubling the exposure halves the rate and halves every bound.
• Confirm the exact upper bound exceeds the normal upper bound at count 12 with the exact width larger, and that at k = 200, T = 4000 the two upper bounds agree within 5 percent. The Garwood interval is not centered, so at count 12 the exact lower bound 0.025836 sits above the normal lower bound 0.021710; the guaranteed widening is on the upper side and in total width, which is the check this leaf asserts (assumption recorded against the wave-39 spec containment bullet, which lists only the upper-side relation in its parenthetical).
• Confirm chi_square_quantile(2, 0.5) reproduces the closed form -2 * ln(0.5) to 1e-9 and that chi2(1, 0.5) is 0.454936.
• Confirm every dict carries exactly the keys rate, lower, upper, method with the documented method string, and that repeated runs are deterministic.
• Confirm ValueError rejection: negative or fractional count, zero or negative exposure, confidence level at or outside (0, 1), df below 1 or fractional, and q at or outside (0, 1).
• Run the contract test offline: python3 scripts/test_poisson_confidence_interval.py (33 tests, deterministic, under 1 second).

Related leaves

• cross-cutting/numerics/confidence-interval-estimation: t and variance intervals for continuous measurements, no count or rate term.
• cross-cutting/numerics/proportion-confidence-interval: Wilson and exact intervals for a success-over-trials fraction, no exposure term.
• cross-cutting/numerics/hypothesis-testing: significance tests that consume the same count data and produce the verdict layer.
• manufacturing-quality/as9100/attribute-control-charts: normal approximation chart limits for attribute data in a different pack and estimator family.

Pitfalls

• Quoting the normal approximation for a small count: the Wald form understates the upper risk badly at low k (12/240 upper 0.0783 against the exact 0.0873) and collapses to a point at k = 0, so the exact Garwood bound is the gate for rare-event claims.
• Forgetting the exposure: a count without the exposure T is meaningless for a rate claim, and both bounds scale as 1 / T, so quoting k = 12 "defects per 240 cycles" as 12 per cycle inflates the rate 240-fold.
• Reading the two-sided split backwards: the lower bound needs the alpha / 2 quantile at df = 2k and the upper bound the 1 - alpha / 2 quantile at df = 2k + 2; swapping the tail sides or the degree-of-freedom shift moves both bounds off the stated level.
• Assuming the exact interval contains the normal interval: the Garwood interval is not centered, and at count 12 its lower bound 0.02584 sits above the normal lower 0.02171 while only the upper bound and total width widen; the honest claim is exactness of coverage, not symmetric containment.
• Treating chi2(2k, alpha / 2) as a table lookup with the wrong tail: chi2(24, 0.025) is 12.401, not the 39.364 upper-tail value, and the misread interval shrinks to nothing on the lower side.
• Hand-rolling the quantile without the survival function: the chi-square quantile needs the inverse of P(df / 2, x / 2), which the in-leaf bisection computes to 1e-9; a rough normal shortcut breaks the small-count anchors.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_poisson_confidence_interval.py

The test covers the spec anchors (exact 12/240 lower 0.02584 and upper 0.08734 within 1e-4, normal 12/240 0.0217 and 0.0783 within 2e-3, zero-count 0/100 upper 0.0369 within 2e-4), chi-square quantile anchors and the df = 2 closed form, rate containment, exposure scaling, the upper-bound widening and the 5 percent convergence at k = 200, dict key and method string contracts, determinism, and ValueError rejection of every non-physical input class.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is the aeronautics numerics reference id per standards-map.yaml; the Poisson interval relations above are standard engineering methodology, summary-only.
• compliance: STANDARDS-REF, gated: false.