Fisher Information and Differentiability for r-Largest Generalized Extreme-Value Models

Abstract

The r largest observations of a block form a dependent random vector whose support moves with all three parameters of the r-largest generalized extreme-value (rGEV) model. For a fixed number r of order statistics, we show that the model is differentiable in quadratic mean if and only if k₀<1/2, where k=−γ is the shape parameter; the threshold is thus the same as for block maxima, because only the largest or the smallest retained order statistic can reach the moving boundary. In this range the Fisher information is positive definite and decomposes into the block-maximum information plus contributions of the lower order statistics, and it equals the negative expected Hessian, which we prove without using any estimator. The model is then locally asymptotically normal, whereas for k₀≥1/2 root-n local asymptotic normality fails. Numerically, using the five largest order statistics instead of the block maximum alone lowers the efficient asymptotic standard error by about 45% for the shape parameter and by $20−−24\%$ for the 100-year return level in heavy-tailed cases (k₀≤−0.2), but by only about 2% for the return level when k₀=0.4.

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BibTeX RIS

BibTeX

@misc{park2026fisher,
  title = {{Fisher Information and Differentiability for $r$-Largest Generalized Extreme-Value Models}},
  author = {Jihong Park and Yire Shin and Jeong-Soo Park},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16136},
  eprint = {2610.16136},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16136},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Fisher Information and Differentiability for r-Largest Generalized Extreme-Value Models
AU  - Jihong Park
AU  - Yire Shin
AU  - Jeong-Soo Park
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16136
UR  - https://arxiv.org/abs/2610.16136
AB  - The r largest observations of a block form a dependent random vector whose support moves with all three parameters of the r-largest generalized extreme-value (rGEV) model. For a fixed number r of order statistics, we show that the model is differentiable in quadratic mean if and only if k_0<1/2, where k=−γ is the shape parameter; the threshold is thus the same as for block maxima, because only the largest or the smallest retained order statistic can reach the moving boundary. In this range the Fisher information is positive definite and decomposes into the block-maximum information plus contributions of the lower order statistics, and it equals the negative expected Hessian, which we prove without using any estimator. The model is then locally asymptotically normal, whereas for k_0≥1/2 root-n local asymptotic normality fails. Numerically, using the five largest order statistics instead of the block maximum alone lowers the efficient asymptotic standard error by about 45% for the shape parameter and by $20−−24\%$ for the 100-year return level in heavy-tailed cases (k_0≤−0.2), but by only about 2% for the return level when k_0=0.4.
ER  -

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