The return-level table as a certificate.

Buoy A, daily maxima, 7,405 points. A fitted distribution is usually a number an optimiser stopped at; here it is a box the Krawczyk operator has proved to contain exactly one stationary point of the likelihood, with the score and its Jacobian evaluated over the whole box in interval arithmetic that rounds outward at every operation. The box is what every later number is computed over, so a return level or a statistic printed to four decimals is an enclosure whose width is stated, not a float's last digits. The exponentiated Weibull's box is the widest — three parameters, one of them a shape that trades against the other two — and it is still 2.229e-7. On the annual maxima its likelihood keeps rising as the shape α runs to infinity and the scale to zero (22 points cannot pin three parameters), so there is no maximum to certify and the ledger says REFUSED rather than printing the optimiser's last iterate.

A² weights the tails, which is why the paper prefers it, and on hourly data it is enormous for any family that mis-shapes the tail: the exponential's 16,356 and the normal's 8,416 on buoy A are not close calls. The exponentiated Weibull's 15.2 is an order of magnitude below the lognormal's 161 on A, and it wins C by 110 to 252; on B the order reverses, lognormal 77.9 against 90.1 — so the paper's "best for high-frequency data" is decided for the exponentiated Weibull twice and against it once, and the daily maxima repeat the same split. As the blocks grow the picture changes, but never to the Weibull: on the monthly maxima the lognormal wins A and B and the exponentiated Weibull wins C; on the annual maxima the lognormal wins all three, with the Weibull second or third. Every one of the twelve choices is DECIDED — no two enclosures overlap — which is a statement about these data and these five families, not about the generalized gamma the paper also fits and this machine cannot yet certify (it needs the digamma function).

◆ marks the family Anderson–Darling prefers. The paper's third finding — that the temporal aggregation moves the return level — is here with a number under it: reading only the preferred family at each block size, the 100-year Hs at buoy A is 10.38 m from one block size and 15.20 m from another. The Weibull, fitted to the hourly series, puts the 100-year wave at 5.26 m on buoy A — 6.5 m below the largest hour already in the record — because its tail is too light for hourly Hs, which is exactly what its A² of 3,496 says. Each return level is the quantile at 1 − b/(T·8766) for a block of b hours, an interval extension over the certified box; the enclosures are 10⁻⁷ m wide or narrower and are printed at their upper end. The width is the arithmetic's; the sampling uncertainty of a 100-year level from twenty years is statistical and is not in it — a certified fit is a certified point estimate, and the page says so.