tails_nov3.py on the seed in §6. Where a claim is a stipulation
rather than a result, it is labelled stipulated.Four sections of fifty leads. Each lead has a number, a single falsifiable sentence, and a disposition: test means the claim can be checked with code or a public table that already exists, and is worth checking now; note means the claim is worth recording but is either a design rule, a reporting convention, or blocked on data nobody has assembled. Nothing is marked with a third tag because nothing here has been tested outside the five simulations in §6.
The leads are deliberately close to the measurements. Where a number appears in a lead it is a threshold that a test would have to hit, not a result. The simulations in §6 are the only places on this page where a number is a measurement, and every one of them is from a synthetic sample.
What this page is for: the questions that decide whether a published tail index means anything. What it is not: a review, a benchmark, or evidence about any real catalogue.
Every lead in this section is about how an estimator behaves when the data it sees is not the data its theory assumes. Truncation here means the largest events are missing (deleted) or capped (censored) — detector saturation, an energy cut-off, or a table that stops at a maximum. The direction of the resulting bias is not the intuitive one, and §6.1 measures it: deleting the top 5% raises the fitted index to 3.3 times its true value, while deleting the bottom 80% does nothing at all.
| # | Lead | |
|---|---|---|
| A1 | Hill's index on Pareto data truncated above at the 95th percentile exceeds the true index by more than a factor of two at fixed k. | test |
| A2 | The bias from upper truncation is upward, because removing the largest events leaves an observed tail lighter than the model rather than heavier. | test |
| A3 | At fixed k, Hill's bias under upper truncation increases monotonically with the truncated fraction. | test |
| A4 | Rescaling k to a fixed fraction of the surviving sample does not remove upper-truncation bias and can increase it. | test |
| A5 | Lower truncation up to the 80th percentile leaves Hill's index within 1% of truth, provided the threshold sits above the cut. | test |
| A6 | A nominal 95% Hill interval covers the truth less than 1% of the time when 5% of the largest events are removed. | test |
| A7 | Intervals built as estimate over root k are anti-conservative exactly where truncation is present, so coverage detects truncation better than the point estimate does. | test |
| A8 | The Pareto maximum-likelihood and Hill estimates coincide at the same k, so reporting both as agreement is one estimator reported twice. | test |
| A9 | The Pickands estimator's standard deviation at k=200, n=4000 is at least three times Hill's on clean Pareto data. | test |
| A10 | The Dekkers–Eenink–de Haan moment estimator is positively biased at k=200 on clean Pareto data. | test |
| A11 | The probability-weighted moment estimator is biased upward by more than 10% at 200 exceedances on clean Pareto data. | test |
| A12 | Probability-weighted moment bias falls roughly as one over the exceedance count, so its small-sample error is a property of m and not of the catalogue. | test |
| A13 | At equal k the spread ordering of the four common estimators is Hill, then moment, then PWM, then Pickands. | test |
| A14 | An estimator ranking measured on clean simulated data bounds the disagreement that any cross-paper comparison can attribute to method. | note |
| A15 | A rise in the fitted index as k falls below about 50 is a bound detector, not a threshold problem. | test |
| A16 | Hill applied to data capped at a saturation value returns an index increasing without limit as k approaches 1. | test |
| A17 | Censoring values at a bound and deleting them produce Hill estimates that differ by less than 10% at the same k. | test |
| A18 | A 5% saturation cut biases the index by at least an order of magnitude more than a 5% random-missingness rate does. | test |
| A19 | Random missingness of 20% leaves Hill unbiased and inflates its standard deviation by about 12%. | test |
| A20 | Inverse-detection-probability weighted Hill removes saturation bias when the detection curve is known. | test |
| A21 | Where detection is a step function, no reweighting recovers the index above the step, so only the sub-step range of k is admissible. | note |
| A22 | A two-piece model with a Pareto body and an atom at the bound recovers the index within 5% at a 5% cut, where Hill errs by more than 200%. | test |
| A23 | The fraction of events sitting exactly at the maximum value is a sufficient diagnostic for saturation. | test |
| A24 | A catalogue whose top value appears more than once should be treated as censored until shown otherwise. | note |
| A25 | Hill's asymptotic standard error is accurate to within 10% for k of 100 or more on clean Pareto data, so it cannot explain the coverage collapse under truncation. | test |
| A26 | Pickands never attains a lower RMSE than Hill for any k between 50 and 800 at n=4000. | test |
| A27 | The moment estimator's finite-sample bias at k=200 has the same sign as the PWM bias. | test |
| A28 | Moment and Hill share a limit, so their difference at fixed k is a finite-sample quantity and should be reported as one. | note |
| A29 | Any estimator comparison at unequal k is uninterpretable, so equal-k comparison should be the default in replication. | note |
| A30 | On lognormal data Hill's index grows without bound as k falls, and no plateau exists at any k. | test |
| A31 | A Hill plateau that is flat over a factor of four in k but 50% from the truth is more dangerous than the absence of a plateau. | note |
| A32 | Bootstrapping the Hill estimate reproduces its bias, so a bootstrap interval cannot detect truncation. | test |
| A33 | Refitting after deleting the largest 1% of events moves the index by more than the bootstrap interval width whenever truncation is present. | test |
| A34 | A leave-the-top-1%-out jackknife is a cheap saturation diagnostic and belongs beside any catalogue index. | note |
| A35 | A generalised-Pareto fit with a free location parameter absorbs part of the truncation bias, at the cost of variance. | test |
| A36 | The corrected maximum-likelihood estimator for a Pareto with known bound is closed-form, so saturation is fixable when the bound is exact. | test |
| A37 | Estimating the bound from the sample maximum makes the corrected estimator biased downward by more than 10% at n=1000. | test |
| A38 | A generalised-Pareto fit to data truncated at the 95th percentile returns a negative shape estimate in more than half of samples. | test |
| A39 | A negative shape estimate on catalogue data is usually evidence of an upper cut rather than evidence against heavy tails. | note |
| A40 | Estimator disagreement that grows with the threshold indicates truncation; disagreement that stays constant indicates sample size. | test |
| A41 | Two estimators at the same k can differ by a factor of three on clean data and by more than ten on truncated data. | test |
| A42 | Hill's estimate of the index is biased upward even when its estimate of the reciprocal is nearly unbiased, so the reporting scale changes the sign of the small-sample bias. | test |
| A43 | Reporting the median rather than the mean removes at least half of the PWM small-sample bias. | test |
| A44 | Any index quoted without k, without the threshold and without the truncation fraction is not reproducible. | note |
| A45 | Under truncation the standard error of the index is smaller than its bias, so precision statements about a truncated index are misleading in the direction of confidence. | test |
| A46 | Reweighting cannot repair a truncated index when the truncation fraction is itself unknown, which is the usual case. | note |
| A47 | An estimator selected because it is stable across thresholds has been selected for insensitivity, not for accuracy. | note |
| A48 | The choice between Hill and PWM moves a catalogue index by more than the physical effects such indices are used to compare, once truncation exceeds 2%. | test |
| A49 | Estimator dependence should be reported as a range, because a single number implies a precision no method delivers on truncated data. | note |
| A50 | Four estimators agreeing on truncated data is evidence that all four are measuring the truncation. | note |
Threshold choice is a fitted parameter that is almost never reported as one. These leads treat selection rules as estimators with their own bias and variance, and ask what a stability plot can and cannot detect. The single measured result in §6.4 is that the rule costs about 20% in standard deviation over a fixed k — small next to the factor-of-four penalty both pay against an oracle k that no analyst has.
| # | Lead | |
|---|---|---|
| A51 | At n=1000, choosing k well matters more than which threshold rule is used: the gap between a fixed k and the oracle exceeds the gap between the rule and the fixed k. | test |
| A52 | The rule 'take the smallest k at which doubling k changes the estimate by less than 5%' inflates the standard deviation by about 20% relative to a fixed k on the same data. | test |
| A53 | No data-driven threshold rule recovers the oracle's precision within a factor of two at n=1000. | test |
| A54 | That rule's chosen k varies by a factor of two across replicates of the same model, so its output is not a property of the data. | test |
| A55 | Apparent flatness over a factor of 1.5 in k is consistent with noise alone at n=1000. | test |
| A56 | Requiring flatness over a factor of four in k would reject more than half of clean samples at n=1000. | test |
| A57 | The plateau width a rule demands controls its bias more than its variance. | note |
| A58 | A threshold chosen by minimising an assumed asymptotic mean squared error is more biased than a fixed rule when the second-order assumption fails. | test |
| A59 | Second-order parameter estimates are too unstable below k=200 to be used for choosing thresholds. | test |
| A60 | The k that is optimal for Hill is not the k that is optimal for PWM, so estimator and threshold cannot be chosen separately. | test |
| A61 | The k that minimises error in the mean of the index is not the k that minimises error in its 95th percentile. | test |
| A62 | Stability of the index across thresholds is a weaker diagnostic than stability of a fixed return level. | test |
| A63 | Return-level stability plots cross more often than Hill plots flatten, so the two disagree about the admissible range. | test |
| A64 | A threshold that passes a stability check on the point estimate can still fail on coverage, and coverage is the quantity that matters. | test |
| A65 | Scanning 30 candidate thresholds at the 5% level produces a false plateau in at least a third of pure-noise samples. | test |
| A66 | An eyeballed plateau is a selection over many k and needs a multiplicity correction before it is quoted. | note |
| A67 | The number of thresholds examined should be reported with any threshold-dependent estimate. | note |
| A68 | Threshold uncertainty inflates the interval of a 100-year return level more than parameter uncertainty does. | test |
| A69 | Bootstrapping over thresholds gives intervals wider than the delta method by a factor of two at n=1000. | test |
| A70 | A threshold selected on the same data used to fit doubles the optimism of the fit. | test |
| A71 | Sample splitting removes threshold-selection optimism at the cost of halving n. | test |
| A72 | Cross-fitted threshold selection recovers at least half of the oracle's advantage at n=2000. | test |
| A73 | Curvature at the small-k end of the Hill plot detects upper truncation at a 5% cut in more than 90% of samples. | test |
| A74 | The second difference of the fitted index over log k separates truncated from clean samples better than plateau width does. | test |
| A75 | A mean residual life plot and a Hill plot disagree about the admissible threshold range in most catalogues. | test |
| A76 | The threshold at which the mean residual life plot becomes linear sits above the threshold at which the Hill plot flattens. | test |
| A77 | Selecting k by minimising the Kolmogorov–Smirnov distance to a fitted generalised Pareto picks k too high, because the test has little power in the tail. | test |
| A78 | A goodness-of-fit test on 100 exceedances has less than 20% power against a 50% error in the index. | test |
| A79 | A rule based only on the exceedance count is more reproducible than a rule based on the shape of the data. | test |
| A80 | The reproducibility of a threshold rule can be measured by the standard deviation of the chosen k across bootstrap samples. | test |
| A81 | Reporting the standard deviation of the chosen threshold is more informative than reporting the threshold. | note |
| A82 | Completeness limits estimated by maximum curvature agree with those from a Hill plateau within 10%. | test |
| A83 | Where the two methods disagree, the curvature method is the wrong one, because it depends on binning. | test |
| A84 | Binning at a 0.1 step creates spurious Hill plateaus wherever the bin width dominates the local spacings. | test |
| A85 | Rounding creates ties that bias the index upward at small k. | test |
| A86 | The tie fraction should be reported with any index computed from rounded data. | note |
| A87 | Threshold stability should be judged jointly on the index and on the 99th percentile, since the two can move in opposite directions. | test |
| A88 | Where they disagree, the percentile is the quantity of interest and should decide the threshold. | note |
| A89 | Stability across time is a stronger check than stability across threshold. | test |
| A90 | A catalogue with a stable threshold profile but a drifting annual index has a detection problem, not a tail problem. | test |
| A91 | Distinguishing annual drift from noise at an index of 1.5 requires at least 10 exceedances per year. | test |
| A92 | In a real catalogue the fitted index is usually monotone in the threshold; in clean simulated data it is not. | test |
| A93 | Monotonicity of the index in the threshold is a better incompleteness flag than the value of the index. | test |
| A94 | In a truncated catalogue, deleting the top 1% of events moves the index more than moving the threshold by a factor of ten. | test |
| A95 | Using the same threshold rule in two catalogues preserves the sign of their index comparison. | test |
| A96 | Using different rules can flip a claimed factor-of-two difference. | test |
| A97 | A pre-registered threshold rule is the only way to make threshold-dependent indices comparable across papers. | note |
| A98 | Pre-registration is rare because the plateau is usually found after the plot is seen. | note |
| A99 | Stability diagnostics cannot separate a power law from a slowly varying regularly varying function over two decades. | test |
| A100 | The same diagnostics can separate them over four decades, so the number of decades spanned is the binding constraint. | test |
A catalogue column that is a function of another column is not an independent measurement, and a row that duplicates another row is not an independent event. Both are catalogue pathologies that look like physics from inside a tail fit. §6.3 shows a single power-law map with a known exponent moving the fitted index from 2.00 to 2.67 while recovering the mapping exponent to three digits — which is the point: the fit is not wrong, it is answering the wrong question.
| # | Lead | |
|---|---|---|
| A101 | A column that is a deterministic function of another carries that function's exponent into the fitted tail index. | test |
| A102 | A derived column related to its source by a power law of exponent beta gives a fitted index of the source index divided by beta. | test |
| A103 | Fitting a physical hypothesis to a derived column tests the mapping exponent, not the physical index. | test |
| A104 | Inverting the map before fitting removes the systematic entirely, so the correction costs nothing once the exponent is known. | test |
| A105 | Recovering the mapping exponent from the two fitted indices is accurate to within 5% in simulation. | test |
| A106 | When the mapping exponent is unknown, it and the physical index are jointly unidentified from a single derived column. | test |
| A107 | Identification requires an independent index or an independent mapping exponent, and never neither. | note |
| A108 | A correlation above 0.999 between two catalogue columns is a derived-column signature. | test |
| A109 | Log-noise of a few per cent hides the signature while leaving the bias in place. | test |
| A110 | A derived column with 5% log-noise still shifts the fitted index by more than 10% when the mapping exponent differs from one by 10%. | test |
| A111 | Duplicating a random 10% of records leaves the point estimate unchanged and narrows every interval by about 5%. | test |
| A112 | Duplicating only the records above the threshold compresses the spacings and biases the index upward. | test |
| A113 | Duplicate detection by exact value matching fails whenever the values are rounded. | test |
| A114 | Duplicate detection on a time-and-position key is more reliable than on value. | test |
| A115 | A catalogue with 20% duplicates has an effective sample size a fifth smaller than reported, so every interval is about 10% too narrow. | test |
| A116 | Double counting inflates the count of extreme events and therefore shortens every return period computed from counts. | test |
| A117 | A return period computed from raw rows is wrong by the duplication rate. | test |
| A118 | Independence fails when one physical event produces several catalogue rows. | test |
| A119 | Fragmentation adds clusters near the top and biases the index upward, the same direction as saturation. | test |
| A120 | Declustering with a fixed window moves the fitted index by 20% as the window goes from one day to ten. | test |
| A121 | A declustering window chosen to maximise fit quality is a fitted parameter and must be counted as one. | note |
| A122 | Independence failures are visible in the lag-one autocorrelation of exceedance times. | test |
| A123 | The autocorrelation that matters for a tail index is the one at the cluster scale, not at the sampling interval. | note |
| A124 | Runs declustering overestimates the number of clusters when events are regularly spaced. | test |
| A125 | Independence in time does not imply independence of magnitudes. | test |
| A126 | Magnitude–time independence can be tested by comparing the index of the first and second halves of each cluster. | test |
| A127 | A monotone transform preserves the tail index only when the transform is regularly varying with index one. | test |
| A128 | Log transforms preserve the index exactly, so a fit on the log scale cannot detect a scale error. | test |
| A129 | A unit error leaves the index unchanged and multiplies every return level by the unit factor. | test |
| A130 | Tail-index claims are therefore immune to unit errors and return-level claims are not. | note |
| A131 | A derived column built as a product of two heavy-tailed inputs inherits the smaller of the two indices. | test |
| A132 | Fitting the product therefore reports the index of the lighter-tailed input, not the physics of the product. | test |
| A133 | Whole-sample correlation misses tail dependence, so independence checks belong on exceedances only. | test |
| A134 | Rank correlation on the upper tail detects dependence that the full-sample correlation reports as absent. | test |
| A135 | A convex map between two columns strengthens tail dependence relative to whole-sample dependence. | test |
| A136 | A concave map weakens it, so the derived column looks lighter-tailed than its source. | test |
| A137 | Excluding records flagged as uncertain is an upper truncation of unknown size that no estimator repairs. | test |
| A138 | Excluding uncertain records moves the index by more than the interval width in most catalogues. | test |
| A139 | The excluded fraction should be reported with the index because it acts as a silent truncation. | note |
| A140 | Two catalogues of the same events disagreeing about the largest event is evidence of a derived or duplicated column. | test |
| A141 | A join on time and position creates duplicates at a rate set by the matching tolerance. | test |
| A142 | Matching tolerance and duplication rate trade off monotonically, so the effective sample size is a function of the join. | test |
| A143 | Every pipeline should report its row count before and after each join. | note |
| A144 | The tail index is more sensitive to duplicates at the top than to missing records at the bottom. | test |
| A145 | De-duplicating by keeping the largest value biases the index downward. | test |
| A146 | De-duplicating by keeping the mean attenuates the top and biases the index upward. | test |
| A147 | The de-duplication rule moves the index by more than the difference between competing physical models. | test |
| A148 | A derived column can be detected by regressing the log of one column on the log of the other and testing the slope against one. | test |
| A149 | That test has useful power only above about 1000 rows, so small catalogues cannot be screened this way. | test |
| A150 | Every index from a catalogue with derived columns should state which column was fitted and which were derived. | note |
Return levels are the deliverable of extreme-value practice and the place where every error above becomes a number in a design document. §6.5 extrapolates a correctly specified model a thousandfold and finds the median almost unbiased while the root-mean-square error exceeds 100%: the point estimate is fine on average and useless individually.
| # | Lead | |
|---|---|---|
| A151 | The median return level extrapolated a thousandfold beyond the sample is within 10% of the truth in simulation. | test |
| A152 | The same estimator's return-level RMSE at that distance exceeds 100%. | test |
| A153 | The 5–95% range of a return level widens by more than a factor of four between the sample range and a thousandfold extrapolation. | test |
| A154 | A return level quoted without a return period and a sample size is not a quantity. | note |
| A155 | Beyond ten times the largest observation, the answer depends more on the estimator than on the data. | test |
| A156 | The 100-year level from a 20-year record is determined mainly by the threshold choice. | test |
| A157 | Doubling the record length reduces return-level RMSE by less than the square root of two, because the threshold must rise. | test |
| A158 | Delta-method intervals for return levels understate the true spread by a factor of two at a 100-year period. | test |
| A159 | Profile-likelihood intervals for return levels are asymmetric and wider on the upper side. | test |
| A160 | The upper end of a return-level interval is the end used in design and the end estimated worst. | note |
| A161 | A fitted generalised Pareto will produce return levels above any physical bound on the quantity, without limit, as the period grows. | test |
| A162 | Where a physical bound exists, a truncated tail model fits the largest periods better than a generalised Pareto. | test |
| A163 | Choosing between a generalised Pareto and a truncated alternative is undecidable from the data below a 100-year period. | test |
| A164 | Return-level sensitivity to the index is the log of the inverse exceedance probability over the index squared, which exceeds one for a 100-year period at an index of 1.5. | test |
| A165 | A 10% error in the index therefore moves a 1000-year level by about half its value at n=1000. | test |
| A166 | Threshold uncertainty dominates a return level once the period exceeds the sample size. | test |
| A167 | Constraining the index to be positive removes the bounded-tail alternative and doubles the return-level bias where a bound exists. | test |
| A168 | Fitting only the top ten exceedances gives return levels with a factor-of-ten spread. | test |
| A169 | Pooling two catalogues halves the return-level RMSE only if their thresholds agree. | test |
| A170 | Pooling catalogues with different thresholds yields a mixture whose fitted index is neither of the two. | test |
| A171 | A mixture of two Pareto tails produces an index that drifts with threshold in a V shape. | test |
| A172 | A V-shaped Hill plot is a mixture signature and should not be reported as threshold instability. | note |
| A173 | Return-level estimates should be validated by leave-one-out on the largest event. | note |
| A174 | Leave-one-out changes a 1000-year level by more than its interval width in most samples at n=1000. | test |
| A175 | The largest observation carries more information about a return level than the next hundred combined. | test |
| A176 | Trimming the largest observation damages a return level more than trimming the smallest half of the sample. | test |
| A177 | A return level that moves by more than 50% when one point is removed should be reported with that fact. | note |
| A178 | Bayesian return levels with a flat prior on the index have wider intervals than frequentist ones at a 1000-year period. | test |
| A179 | A flat prior on the index gives an improper predictive upper bound once the index exceeds one. | test |
| A180 | Empirical plotting positions cannot arbitrate between fitted models because they do not extrapolate. | test |
| A181 | Reporting the empirical and the fitted return level side by side exposes extrapolation error at zero cost. | note |
| A182 | Most return-period claims about rare events are threshold claims restated. | note |
| A183 | For bounded data a generalised Pareto fit returns a negative index and an upper endpoint, and the endpoint is the more stable quantity. | test |
| A184 | The endpoint estimate is within 20% of a true bound that sits ten times above the threshold. | test |
| A185 | Extrapolation accuracy depends on the exceedance count, not on n, so a larger catalogue with the same k buys nothing. | test |
| A186 | Doubling k halves the return-level variance and doubles the bias, and the optimum sits where they cross. | test |
| A187 | Return levels are invariant to the units of the data, so a unit error is invisible in a stability plot. | test |
| A188 | Threshold uncertainty dominates at short periods and shape uncertainty dominates at long ones. | test |
| A189 | A one-in-N claim requires N to be shorter than the record for the claim to be checkable. | note |
| A190 | The one-in-N framing hides the model choice, because N is defined by the fit. | note |
| A191 | Replacing a fitted tail with the empirical tail turns a one-in-100 claim into a one-in-20 claim in a 100-event sample. | test |
| A192 | Records set early in a catalogue are less likely to be exceeded than records set late if detection improved. | test |
| A193 | A catalogue with improving detection produces an index that appears to fall with time, with no physical change. | test |
| A194 | Return levels should be computed per detection era and combined, never pooled. | note |
| A195 | Era-wise pooling needs at least 30 exceedances per era, which most catalogues lack. | test |
| A196 | The number of independent exceedances should replace n in every return-level formula. | note |
| A197 | Using raw counts instead of declustered counts makes return periods too short by the clustering factor. | test |
| A198 | A return-level interval that ignores threshold uncertainty is too narrow by more than a factor of two at long periods. | test |
| A199 | Every quantity in this section is a function of the threshold, so no return-level claim is complete without one. | note |
| A200 | A tail index and a return level from the same fit can disagree about stability, and the return level is the one that matters. | note |
Five simulations, all on synthetic data, all in incoprea_nov3/code/tails_nov3.py, with results
in incoprea_nov3/results/tails_nov3.json. Sample: numpy.random.default_rng(20261103),
400 replicates per cell, numpy 2.5.4. Data are Pareto with survival x−α on
[1,∞) unless stated; the Hill statistic here estimates ξ = 1/α and the tables report
1/H, so every number is on the α scale. A first version of the script reported H itself as α and
used a probability-weighted-moment coefficient derived for the wrong sign convention; the recovery table below
is the check that caught both, and the numbers in this section are from the corrected run.
Recovery check. On clean Pareto data with α=1.5, n=20000, both estimators must return 1.5 or nothing below is interpretable.
| Setting | Hill α | PWM α | PWM undefined |
|---|---|---|---|
| 100 | 1.528 | 1.904 | 0 |
| 400 | 1.503 | 1.613 | 0 |
| 1600 | 1.497 | 1.555 | 0 |
| GPD check, ξ=0.25 | 4.000 | 0.249 | 0 |
| GPD check, ξ=0.50 | 2.000 | 0.494 | 0 |
Simulation. Pareto(α=1.5), n=4000, k=200 held fixed. "Truncated above" deletes the largest fraction f of the sample — what a saturating detector or a table with a maximum value does. Coverage is the fraction of replicates in which the nominal 95% interval α̂ ± 1.96 α̂/√k contains 1.5.
| Truncated above | n kept | Hill α | Bias | Hill s.d. | 95% coverage | Hill α, k rescaled | PWM median ξ | PWM valid |
|---|---|---|---|---|---|---|---|---|
| 0.00 | 4000 | 1.511 | +0.7% | 0.109 | 0.963 | 1.511 | 0.596 | 1.00 |
| 0.05 | 3800 | 4.918 | +227.8% | 0.390 | 0.000 | 5.090 | -0.520 | 0.00 |
| 0.10 | 3600 | 7.947 | +429.8% | 0.594 | 0.000 | 8.641 | -0.697 | 0.00 |
| 0.20 | 3200 | 13.988 | +832.5% | 1.163 | 0.000 | 17.070 | -0.828 | 0.00 |
| 0.30 | 2800 | 20.104 | +1240.3% | 1.601 | 0.000 | 27.857 | -0.866 | 0.00 |
| 0.40 | 2400 | 26.156 | +1643.7% | 2.054 | 0.000 | 42.453 | -0.902 | 0.00 |
| 0.50 | 2000 | 32.080 | +2038.7% | 2.727 | 0.000 | 62.017 | -0.911 | 0.00 |
Reading. At f=0 the estimator is fine: mean 1.511 against 1.5, coverage 0.963, and the standard error is honest (s.d. 0.109 against the formula's 0.106). At f=0.05 the mean is 4.918 — 3.3 times the truth — and the coverage is 0.000. The error is bias, not noise: the RMSE at that point is 3.44, about nine times the standard deviation of 0.390. Rescaling k to 5% of the survivors makes it worse (5.090), because the survivors' top order statistics are compressed against the bound whichever k is chosen.
The direction is the opposite of the intuitive one. Removing the biggest events does not
make the observed tail heavier; it makes it lighter, so the fitted index climbs. The PWM estimator returns a
negative median ξ (-0.52 at f=0.05) and a valid positive index in 0% of
replicates: it is not biased, it is correctly reporting that no heavy tail remains. That is a diagnosis, and
reported as nan it looks like a failure.
Lower truncation is harmless and that asymmetry is the useful part. Deleting the smallest 80% of events, with the threshold still above the cut:
| Truncated below | n kept | Hill α | Bias | Hill s.d. | 95% coverage |
|---|---|---|---|---|---|
| 0.00 | 4000 | 1.509 | +0.6% | 0.103 | 0.953 |
| 0.20 | 3200 | 1.508 | +0.6% | 0.105 | 0.960 |
| 0.50 | 2000 | 1.506 | +0.4% | 0.108 | 0.938 |
| 0.80 | 800 | 1.501 | +0.1% | 0.105 | 0.950 |
Bias stays at +0.6% and coverage at 0.960. Incompleteness at the bottom is a threshold-placement problem; incompleteness at the top is an estimator problem. The practical reading: a catalogue that is incomplete for small events is safe once the threshold clears the completeness limit, and a catalogue that saturates at the top cannot be fixed by any choice of threshold or estimator, only by modelling the bound.
Simulation. Pareto(α=1.5), k fixed at 10% of n, 400 replicates. "Off by >50%" is the fraction of replicates whose estimate is more than 50% from the truth. Medians are reported beside means because the PWM distribution has no usable variance below about 100 exceedances.
| n | k | Hill median α | Hill s.d. | PWM median α | PWM s.d. | PWM 10–90% | s.d. ratio | PWM off by >50% |
|---|---|---|---|---|---|---|---|---|
| 100 | 20 | 1.527 | 0.363 | 2.106 | 23.866 | 1.35–5.82 | 65.76 | 0.468 |
| 250 | 25 | 1.526 | 0.289 | 2.167 | 6.193 | 1.33–4.92 | 21.46 | 0.468 |
| 500 | 50 | 1.512 | 0.234 | 1.890 | 1.526 | 1.29–3.25 | 6.52 | 0.342 |
| 1000 | 100 | 1.500 | 0.143 | 1.760 | 0.571 | 1.33–2.65 | 4.00 | 0.193 |
| 2500 | 250 | 1.510 | 0.103 | 1.676 | 0.285 | 1.36–2.04 | 2.77 | 0.028 |
| 5000 | 500 | 1.498 | 0.070 | 1.639 | 0.218 | 1.38–1.92 | 3.11 | 0.003 |
| 20000 | 2000 | 1.501 | 0.031 | 1.565 | 0.130 | 1.40–1.70 | 4.13 | 0.000 |
Reading. Hill is the maximum-likelihood estimator for a Pareto tail and it shows: at n=1000 its median is 1.500 with s.d. 0.143, while PWM's median is 1.760 with s.d. 0.571, a ratio of 4.0. PWM is also biased upward at every sample size tested — +26.4% at n=1000 — and 19% of its estimates are more than 50% off the truth at that size. The bias falls roughly as one over the exceedance count, which is why the same estimator looks stable on a large catalogue and unstable on a small one.
This contradicts how I read the earlier catalogue result. A probability-weighted moment estimate that is stable to three digits across a 20× threshold range is not thereby the more accurate estimator; it is the less responsive one. Stability and efficiency are being traded, and a stability plot alone cannot tell which side of the trade a reported number sits on.
Simulation. X is Pareto(α=2), n=3000, k=150. The catalogue column is Y = Xβ · exp(ε) with ε ~ Normal(0, 0.05²) — a total energy reconstructed from a radiated energy through an efficiency that drifts with size. For Y the tail exponent is α/β exactly, so the fitted index moves while the fit stays correct.
| Mapping exponent β | Predicted α/β | Fitted α | Bias vs α=2 | Implied β |
|---|---|---|---|---|
| 0.60 | 3.333 | 3.356 | +67.8% | 0.596 |
| 0.75 | 2.667 | 2.673 | +33.7% | 0.748 |
| 0.90 | 2.222 | 2.240 | +12.0% | 0.893 |
| 1.00 | 2.000 | 2.020 | +1.0% | 0.990 |
| 1.25 | 1.600 | 1.612 | -19.4% | 1.241 |
Reading. At β=1 the recovery is clean (+1.0%), so the machinery is not adding the effect. At β=0.75 the fitted index is 2.673 against a predicted 2.667, and the implied exponent 0.748 recovers β to three digits. Every intermediate β does the same. The consequence is not that the fit is wrong; it is that a hypothesis stated in terms of the source quantity has been tested against a column that answers a different question, and the answer is deterministic rather than noisy. Inverting the map before fitting removes all of it, which makes this the cheapest fix on the page and the one least likely to be done, because the derived column is usually the one in the table.
Simulation. Pareto(α=1.5), n=1000, 400 replicates, same data for all three columns. The rule takes the smallest k in [30, 300] at which doubling k changes the estimate by less than 5% — the rule one writes when judging a Hill plot by eye. The oracle takes the k closest to the truth, which no analyst has, and is there to size the gap.
| Threshold rule | Mean α | s.d. | RMSE | Bias |
|---|---|---|---|---|
| Fixed k = 100 | 1.517 | 0.148 | 0.149 | +1.2% |
| Stability rule | 1.519 | 0.178 | 0.179 | +1.3% |
| Oracle k (unavailable in practice) | 1.501 | 0.034 | 0.034 | +0.1% |
Reading. The rule is worse than a fixed k by 21% in s.d. (0.178 against 0.148) and no better in bias, and its chosen k runs from 30 to 65 with a median of 40 across replicates of an identical model. Both, however, sit a factor of about 4.4 away from the oracle's s.d. of 0.034. The ordering matters for practice: at this sample size the choice of k dominates the choice of rule, so effort spent on a better automatic selector is spent in the wrong place.
I expected the rule to be much worse than a fixed k. It is only 21% worse, which is the size of an effect that would vanish in most comparisons across catalogues.
Simulation. Pareto(α=1.5), n=1000 observations, Hill fit at k=100, extrapolated to the level exceeded once per T blocks of n observations, so p = 1/(nT). The model is correctly specified, which makes these the best numbers this method can produce.
| Return period (blocks) | Exceedance prob. | True level | Median estimate | Bias of median | RMSE | 95th/5th pct. | Level shift for +10% α |
|---|---|---|---|---|---|---|---|
| 1 | 1.0e-03 | 100 | 98.93 | -1.1% | 32.5% | 2.68 | -24% |
| 10 | 1.0e-04 | 464.2 | 429.4 | -7.5% | 56.1% | 4.31 | -34% |
| 100 | 1.0e-05 | 2154 | 2121 | -1.5% | 127.3% | 8.71 | -43% |
| 1000 | 1.0e-06 | 1e+04 | 9676 | -3.2% | 115.9% | 12.43 | -50% |
Reading. The median is close to the truth at every distance — -1.1% at T=1 and -3.2% at T=1000 — so the estimator is not systematically wrong. The spread is another matter: RMSE rises from 32% to 116%, and the 5–95% range goes from a factor of 2.68 to 12.43. The last column is the sensitivity that matters for practice: a 10% error in α moves the 1000-block level by -50%. Since §6.1 shows truncation moving the index by hundreds of per cent, the index error, not the extrapolation arithmetic, is what makes a long-period return level unreliable.
My expectation was that bias would dominate at long extrapolation distances. It does not: the median stays within 8% while the spread grows fourfold. The failure mode is not a wrong number, it is a number with no purchase — and it will be quoted without its interval.
Simulation. Pareto(α=1.5), n=4000, 400 replicates, no truncation and no derived column. This is the best case for all four methods, so any disagreement here is method and not data.
| k | Estimator | Mean α | Bias | s.d. | RMSE |
|---|---|---|---|---|---|
| 100 | Hill | 1.519 | +1.3% | 0.158 | 0.159 |
| 100 | Pickands | 1.690 | +12.6% | 0.743 | 0.766 |
| 100 | Moment | 1.654 | +10.3% | 0.422 | 0.449 |
| 100 | PWM | 1.901 | +26.7% | 0.659 | 0.771 |
| 200 | Hill | 1.507 | +0.5% | 0.109 | 0.109 |
| 200 | Pickands | 1.551 | +3.4% | 0.383 | 0.386 |
| 200 | Moment | 1.560 | +4.0% | 0.221 | 0.229 |
| 200 | PWM | 1.714 | +14.3% | 0.346 | 0.407 |
| 400 | Hill | 1.505 | +0.3% | 0.075 | 0.075 |
| 400 | Pickands | 1.557 | +3.8% | 0.281 | 0.286 |
| 400 | Moment | 1.528 | +1.8% | 0.141 | 0.143 |
| 400 | PWM | 1.636 | +9.1% | 0.241 | 0.276 |
Reading. At k=200: Hill 1.507 (s.d. 0.109), moment 1.560 (0.221), PWM 1.714 (0.346), Pickands 1.551 (0.383). The ordering is stable across k=100, 200 and 400, and all three non-Hill estimators are biased upward at every k tested. Hill is the maximum-likelihood estimator under the model, so this is what theory predicts; the useful part is the size. Pickands' s.d. is 3.5 times Hill's at k=200, which means a table of Pickands estimates from 100 events carries an uncertainty comparable to the factor between competing physical models.
The moment estimator is the surprise: asymptotically it is a different estimator, but at these k values it sits closer to Hill than PWM does in both bias and spread.
Two hundred leads, of which 159 are marked test and 41 note. The tags are a priority claim, not a quality claim: "test" means a check is cheap and would change what I do next, "note" means it would not, yet.
| Section | Leads | test | note | Numbering |
|---|---|---|---|---|
| Section A · Estimator behaviour on truncated data | 50 | 37 | 13 | A1–A50 |
| Section B · Threshold selection and stability diagnostics | 50 | 42 | 8 | A51–A100 |
| Section C · Derived columns, double counting and independence failures | 50 | 43 | 7 | A101–A150 |
| Section D · Extreme-value practice: return levels, extrapolation, and what breaks them | 50 | 37 | 13 | A151–A200 |
| Total | 200 | 159 | 41 | A1–A200 |
If one line survives: a tail index from truncated data is wrong in the direction of looking lighter, by a factor that grows with the truncation, and no threshold or estimator choice recovers it — while the estimator that looks most stable across thresholds is the one least able to tell you that this has happened.