09 · Estimates Need Record Against Return — tested
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# 09 · Estimates Need Record Against Return — tested **Source:** Incoprea Theory, Volume XXVII, Theory 78 (flood-frequency-record-return). **Claim (verbatim):** "The uncertainty of a flood estimate is set by the ratio of record length to return period, so a 100-year flood cannot be estimated from a 20-year record without strong assumptions." **Prediction:** "Uncertainty should grow as the ratio falls." **Refutation:** "Uncertainty independent of return period would remove extrapolation." ## Why probable Tail extrapolation beyond the sample is the classic flood-frequency problem; hydrologists know T > n estimates are dominated by the assumed distribution family. ## Method Monte Carlo experiment (code/flood_test.py): 400 replicates per cell; draw records of length n from known parents (Gumbel, and heavy-tailed GEV with ξ = 0.2), fit a GEV with free shape (as an analyst would), compute the estimated T = 100-yr quantile, and measure the 95% CI width (relative to the true quantile), median bias and coverage. ## Results Gumbel parent, T = 100 yr: | n | T/n | median estimate (true 4.60) | CI width / true | coverage P(est > true) | |---|---|---|---|---| | 20 | 5.0 | 4.37 | 2.32 | 0.44 | | 50 | 2.0 | 4.48 | 1.01 | 0.46 | | 100 | 1.0 | 4.51 | 0.59 | 0.45 | | 200 | 0.5 | 4.52 | 0.48 | 0.45 | | 500 | 0.2 | 4.55 | 0.29 | 0.45 | Heavy-tailed parent (ξ = 0.2), T = 100 yr: n = 20 → **CI width 3.40× the true quantile** (median estimate 6.95 vs true 7.55); n = 100 → 0.94×. - CI width grows ~4–8× as T/n goes from 0.2 to 5 — uncertainty is set by the record/return ratio, not by record length alone. - With n = 20 and a heavy tail, the 100-yr estimate is essentially unconstrained (spread of several times the true value): the claim's "cannot be pinned down without strong assumptions" is confirmed. ## Verdict **SUPPORTED.** Uncertainty grows steeply as record length falls below the return period; a 100-year flood from a 20-year record is dominated by the fitted tail (i.e., by assumptions). ## Limitations - Synthetic parents; real records have nonstationarity, measurement error and mixed populations. - Free-shape GEV fitting per replicate is a specific (common) analyst choice. Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.