Parametric and nonparametric probability distribution estimators of sample maximum

Taku Moriyama

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Parametric and nonparametric probability distribution estimators of sample maximum

Taku Moriyama

Statistics Theory · Submitted 6 Nov 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The numerical performances of the parametric fitting estimator and the nonparametric estimator are compared, which shows that the nonparametric estimator performs better, especially for the extreme value index close to zero.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.40 · human participants 0.02 · controlled comparison 0.56 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.96 · mentions code or data availability 0.02.

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arXiv listing abstract v5. Results and the main claim are underlined; show every role.

Extreme value theory has constructed asymptotic properties of the sample maximum. This study concerns probability distribution estimation of the sample maximum. The traditional approach is parametric fitting to the limiting distribution -- the generalized extreme value distribution; however, the model in non-limiting cases is misspecified to a certain extent. We propose a plug-in type of nonparametric estimator that does not need model specification. Asymptotic properties of the distribution estimator are derived. The simulation study numerically investigates the relative performance in finite-sample cases. This study assumes that the underlying distribution of the original sample belongs to one of the Hall class, the Weibull class or the bounded class, whose types of the limiting distributions are all different: the Frechet, Gumbel or Weibull. It is proven that the convergence rate of the parametric fitting estimator depends on both the extreme value index and the second-order parameter and gets slower as the extreme value index tends to zero. On the other hand, the rate of the nonparametric estimator is proven to be independent of the extreme value index under certain conditions. The numerical performances of the parametric fitting estimator and the nonparametric estimator are compared, which shows that the nonparametric estimator performs better, especially for the extreme value index close to zero. Finally, we report two real case studies: the Potomac River peak stream flow (cfs) data and the Danish Fire Insurance data.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Extreme value theory has constructed asymptotic properties of the sample maximum.	background 1.00	0.04	0.02	asserted 1.00
s1	This study concerns probability distribution estimation of the sample maximum.	contribution 0.68	0.49	0.03	asserted 0.93
s2	The traditional approach is parametric fitting to the limiting distribution -- the generalized extreme value distribution; however, the model in non-limiting cases is misspecified to a certain extent.	background 0.59	0.08	0.04	asserted 0.91
s3	We propose a plug-in type of nonparametric estimator that does not need model specification.	contribution 0.78	0.91	0.10	asserted 1.00
s4	Asymptotic properties of the distribution estimator are derived.	contribution 0.57	0.92	0.04	asserted 0.99
s5	The simulation study numerically investigates the relative performance in finite-sample cases.	method 0.93	0.92	0.03	asserted 0.92
s6	This study assumes that the underlying distribution of the original sample belongs to one of the Hall class, the Weibull class or the bounded class, whose types of the limiting distributions are all different: the Frechet, Gumbel or Weibull.	method 0.73	0.39	0.03	asserted 1.00
s7	It is proven that the convergence rate of the parametric fitting estimator depends on both the extreme value index and the second-order parameter and gets slower as the extreme value index tends to zero.	result 0.99	0.86	0.12	asserted 1.00
s8	On the other hand, the rate of the nonparametric estimator is proven to be independent of the extreme value index under certain conditions.	result 0.99	0.92	0.17	asserted 1.00
s9	The numerical performances of the parametric fitting estimator and the nonparametric estimator are compared, which shows that the nonparametric estimator performs better, especially for the extreme value index close to zero.	result 0.99	0.95	0.74	asserted 1.00
s10	Finally, we report two real case studies: the Potomac River peak stream flow (cfs) data and the Danish Fire Insurance data.	contribution 0.65	0.92	0.03	asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.93 · evidence from proof or derivation 0.79.

What was examined, and what was not

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arXiv listing abstract v5

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references and citing sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Extreme value theory has constructed asymptotic properties of the sample maximum. This study concerns probability distribution estimation of the sample maximum. The traditional approach is parametric fitting to the limiting distribution -- the generalized extreme value distribution; however, the model in non-limiting cases is misspecified to a certain extent. We propose a plug-in type of nonparametric estimator that does not need model specification. Asymptotic properties of the distribution estimator are derived. The simulation study numerically investigates the relative performance in finite-sample cases. This study assumes that the underlying distribution of the original sample belongs to one of the Hall class, the Weibull class or the bounded class, whose types of the limiting distributions are all different: the Frechet, Gumbel or Weibull. It is proven that the convergence rate of the parametric fitting estimator depends on both the extreme value index and the second-order parameter and gets slower as the extreme value index tends to zero. On the other hand, the rate of the nonparametric estimator is proven to be independent of the extreme value index under certain conditions. The numerical performances of the parametric fitting estimator and the nonparametric estimator are compared, which shows that the nonparametric estimator performs better, especially for the extreme value index close to zero. Finally, we report two real case studies: the Potomac River peak stream flow (cfs) data and the Danish Fire Insurance data.

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Compares with it 1 paper

These results are not preferable as seen from comparing Table 2 with the MISE values given in [ref] .

A semiparametric probability distribution estimator of sample maximums · Simulation study · 0.98

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