pith. sign in

arxiv: 1907.02175 · v1 · pith:NMZ6VCMMnew · submitted 2019-07-04 · 📊 stat.AP

Bayesian analysis of extreme values in economic indexes and climate data: Simulation and application

Ali Reza Fotouhi This is my paper

Pith reviewed 2026-05-25 09:15 UTC · model grok-4.3

classification 📊 stat.AP
keywords extreme value theoryBayesian inferencemixed modelsrandom effectsreturn levelValue-at-Riskclimate dataeconomic indexes
0
0 comments X

The pith

Informative priors from past data reduce bias in estimates of return levels and risk measures for extreme values.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Bayesian approach to mixed modeling of extreme values that incorporates random effects to handle heterogeneity across series or time periods. Simulations show that using posterior distributions from earlier data as informative priors produces lower bias in key quantities than uninformative priors. The same pattern appears when the method is applied to climate records and returns on economic indexes. A reader would care because climate and financial series are often large, incomplete, and arrive sequentially, so retaining historical information improves forecasts of rare events.

Core claim

By modeling parameters of extreme value distributions as random effects and using Bayesian updating to convert information from past data into informative priors, the mixed model produces return level estimates under the block maxima method and Value-at-Risk and Expected Shortfall under the peaks-over-threshold method that show less bias than those obtained with uninformative priors, both in simulated heterogeneous data and in direct applications to climate and economic time series.

What carries the argument

Bayesian mixed model for extreme value distributions in which posterior distributions from past data become informative priors for subsequent data, with random effects capturing heterogeneity.

If this is right

  • Random-effects modeling of extremes yields more reliable parameter estimates than standard non-mixed models whenever heterogeneity is present.
  • Informative priors obtained from historical data improve accuracy for return levels, Value-at-Risk, and Expected Shortfall relative to uninformative priors.
  • The method preserves information across periods, which is useful when data sets are massive and portions may be missing.
  • Simulation and real-data results both confirm reduced bias under the Bayesian mixed approach.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sequential updating structure could support real-time revision of risk measures as new extremes are observed.
  • The same prior-transfer technique may apply to other sequential extreme-event domains such as insurance losses or environmental monitoring.
  • If the random-effects assumption fails to capture important forms of heterogeneity, the reported bias reductions may not hold in practice.

Load-bearing premise

Random effects adequately capture the heterogeneity present in the extreme-value data and the simulation designs match the conditions of real climate and economic series.

What would settle it

A direct comparison, on held-out future observations, of the bias in return level estimates obtained when informative priors derived from earlier data are used versus when uninformative priors are used, under data that exhibits heterogeneity not captured by the random effects.

Figures

Figures reproduced from arXiv: 1907.02175 by Ali Reza Fotouhi.

Figure 6
Figure 6. Figure 6: shows the quantile and return level plots of applying GEV random e [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
read the original abstract

Mixed modeling of extreme values and random effects is relatively unexplored topic. Computational difficulties in using the maximum likelihood method for mixed models and the fact that maximum likelihood method uses available data and does not use the prior information motivate us to use Bayesian method. Our simulation studies indicate that random effects modeling produces more reliable estimates when heterogeneity is present. The application of the proposed model to the climate data and return values of some economic indexes reveals the same pattern as the simulation results and confirms the usefulness of mixed modeling of random effects and extremes. As the nature of climate and economic data are massive and there is always a possibility of missing a considerable part of data, saving the information included in past data is useful. Our simulation studies and applications show the benefit of Bayesian method to save the information from the past data into the posterior distributions of the parameters to be used as informative prior distributions to fit the future data. We show that informative prior distributions obtained from the past data help to estimate the return level in Block Maxima method and Value-at-Risk and Expected Shortfall in Peak Over Threshold method with less bias than using uninformative prior distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes Bayesian mixed models with random effects for extreme value analysis of climate and economic data, using Block Maxima for return levels and Peak Over Threshold for VaR/ES. It claims that informative priors derived from past data reduce bias relative to uninformative priors, that random-effects modeling improves reliability under heterogeneity, and that both simulation studies and real-data applications confirm these benefits.

Significance. If the bias-reduction and heterogeneity-handling claims hold under realistic conditions, the work would provide a practical Bayesian framework for sequential extreme-value modeling that reuses historical information, which is valuable for large, incomplete climate and financial datasets. The emphasis on saving past information in posteriors for future priors is a clear applied strength.

major comments (2)
  1. [Simulation studies] Simulation studies section: the data-generating processes are not shown to incorporate non-stationarity, trends, or long-range dependence; if they assume only exchangeable blocks or simple random intercepts, the reported bias reductions cannot be taken as evidence that the method generalizes to the climate and economic series described in the applications.
  2. [Applications] Applications section: the statement that real-data results 'reveal the same pattern' is presented without quantitative calibration checks (e.g., posterior predictive coverage under observed trends or regime shifts), leaving the central claim that informative priors yield lower bias unsupported for the target data types.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the form of the random-effects distribution and the precise definition of the informative priors (e.g., which parameters receive the past-data posterior as prior).
  2. [Methods] Notation for the mixed-model likelihood and the Block Maxima/POT return-level expressions should be unified across sections to avoid ambiguity when comparing bias results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments point by point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: Simulation studies section: the data-generating processes are not shown to incorporate non-stationarity, trends, or long-range dependence; if they assume only exchangeable blocks or simple random intercepts, the reported bias reductions cannot be taken as evidence that the method generalizes to the climate and economic series described in the applications.

    Authors: We agree that the simulation designs focus on exchangeable blocks with random intercepts to isolate the effect of heterogeneity and the benefit of informative priors derived from past data. These DGPs do not incorporate non-stationarity, trends, or long-range dependence, so the reported bias reductions cannot be interpreted as direct evidence that the method automatically generalizes to every feature of the target series. The simulations serve to demonstrate the advantage of the mixed-model approach under controlled heterogeneity; the applications on climate and economic indexes provide the complementary evidence on real data. We will revise the simulation section to state this scope explicitly and add a brief discussion of how the random-effects framework could be extended (e.g., via time-varying random effects) to accommodate non-stationarity. revision: yes

  2. Referee: Applications section: the statement that real-data results 'reveal the same pattern' is presented without quantitative calibration checks (e.g., posterior predictive coverage under observed trends or regime shifts), leaving the central claim that informative priors yield lower bias unsupported for the target data types.

    Authors: We acknowledge that the applications section relies on qualitative comparison of point estimates and does not report formal posterior predictive checks or coverage diagnostics under the observed trends and possible regime shifts. While the pattern of reduced bias with informative priors is visible in the reported return levels, VaR, and ES, the absence of these quantitative calibrations leaves the claim less strongly supported than it could be. We will add posterior predictive coverage assessments and, where feasible, checks that account for trends in the revised applications section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on external simulations and data applications

full rationale

The paper's central claims—that random-effects mixed models yield more reliable extreme-value estimates and that informative priors derived from past data reduce bias in return levels, VaR, and ES—are supported by simulation studies comparing model variants and by applications to climate and economic index data. No equations or steps in the provided abstract reduce a prediction or fitted quantity to its own inputs by construction, nor does the argument rely on self-citations whose content is unverified. The methodology follows standard Bayesian hierarchical modeling for extremes without importing uniqueness theorems or ansatzes from the author's prior work. The derivation chain is therefore self-contained against the simulation benchmarks and external datasets.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit model equations or parameter lists; the approach implicitly relies on standard extreme-value theory assumptions without introducing new free parameters, axioms, or entities visible at this level of detail.

axioms (1)
  • domain assumption Generalized extreme value distribution adequately describes block maxima and generalized Pareto distribution describes exceedances over thresholds
    Implicit in the use of Block Maxima and Peak Over Threshold methods mentioned in the abstract.

pith-pipeline@v0.9.0 · 5722 in / 1209 out tokens · 24009 ms · 2026-05-25T09:15:10.108697+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    Reiss, R. and D. Thomas, M., Statistical analysis of extreme values: with applications to insurance, finance, hydrology and other fields London : Springer [ebook], 2007

    work page 2007

  2. [2]

    Trapin, L., Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review

    Bee, M. Trapin, L., Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review. Risks, Vol 6, Iss 2, 2018

    work page 2018

  3. [3]

    M, KÄellezi

    Gilli. M, KÄellezi. An Application of Extreme Value Theory for Measuring Financial Risk. Computational Economics 27(1), 1-23, 2006

    work page 2006

  4. [4]

    and Muraviev, R., An extreme-value theory approximation scheme in reinsurance- linked securities

    Aviv, R. and Muraviev, R., An extreme-value theory approximation scheme in reinsurance- linked securities. Astin Bulletin, Vol. 48 Issue 3, p1157-1173, Sep 2018

    work page 2018

  5. [5]

    Szatata, L

    Kuzminski, L. Szatata, L. and Zwozdziak, J. Measuning aquatic environments as a tool for flood risk management in terms of climate changes dynamics. Polish Journal of environmental studies. Vol. 27 Issue 4, p1583-1592, 2018

    work page 2018

  6. [6]

    Flood risk under future climate in data sparse regions: Linking extreme value models and flood generating processes

    Tramblay, Y; Amoussou, E; Dorigo, W; Mahé, G. Flood risk under future climate in data sparse regions: Linking extreme value models and flood generating processes. Journal of Hydrology. 519 Part A:549-558 Part A, 27 November 2014

    work page 2014

  7. [7]

    A comparative review of generalizations of the Gumbel extreme value distribution with an application to wind speed data

    Pinheiro, E. C.; Ferrari, S. L. P. A comparative review of generalizations of the Gumbel extreme value distribution with an application to wind speed data. Working paper in http://arxiv.org/abs/1502.02708 34

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    G.; Sannasiraj, S

    Polnikov, V. G.; Sannasiraj, S. A.; Satish, S.; Pogarskii, F. A.; Sundar, V. Estimation of extreme wind speeds and wave heights along the regional waters of India. In Ocean Engineering, 146:170-177, Dec 2017

    work page 2017

  9. [9]

    de Haan, L.; Ferreira, A. F. Extreme value theory, An Introduction. Springer series in operations research and financial engineering, 2006

    work page 2006

  10. [10]

    Extreme Value Theory in Engineering, Elsevier, 1988

    Enrique C. Extreme Value Theory in Engineering, Elsevier, 1988

    work page 1988

  11. [11]

    and Tippett, L

    Fisher, R. and Tippett, L. H. C. Limiting forms of the frequency distribution of largest or smallest member of a sample. Proceedings of the Cambridge philosophical society, 24:180-190, 1928

    work page 1928

  12. [12]

    Gnedenko, B. V. Sur la distribution limite du terme d'une serie aleatoire. Annals of Mathematics, 44:423-453, 1943

    work page 1943

  13. [13]

    Pickands, J. I. Statistical inference using extreme value order statistics. Annals of Statistics, 3:119-131, 1975

    work page 1975

  14. [14]

    Balkema, A. A. and de Haan, L. Residual lifetime at great age. Annals of Probability, 2:792-804, 1974

    work page 1974

  15. [15]

    Mixed models: Theory and application with R, second edition, Wiley series in probability and statistics

    Demidenko, E. Mixed models: Theory and application with R, second edition, Wiley series in probability and statistics. 2013

    work page 2013

  16. [16]

    Theory of probability, third edition

    Jeffreys, H. Theory of probability, third edition. Oxford University Press, London, 1961

    work page 1961

  17. [17]

    Bayesian inference in statistical analysis, John Wiley and Sons New York, 1973

    Box, G.E.P and Tiao, G.C. Bayesian inference in statistical analysis, John Wiley and Sons New York, 1973

    work page 1973

  18. [18]

    and Hanson, T

    Christensen, R.; Johnson, W, ; Branscum, A. ; and Hanson, T. E. Bayesian ideas and data analysis. CRS Press, Taylor & Francis Group, 2011

    work page 2011

  19. [19]

    and Ulam, S

    Metropolis, N. and Ulam, S. The Monte Carlo Method, Journal of the American Statistical Association, 44, 1949

    work page 1949

  20. [20]

    W., Rosenbluth, M

    Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., Equation of State Calculations by Fast Computing Machines, Journal of Chemical Physics, 21, 1087–1092, 1953

    work page 1953