pith. sign in

arxiv: 2509.15359 · v2 · submitted 2025-09-18 · 📊 stat.ME

Bayesian Mixture Models for Heterogeneous Extremes

Viviana Carcaiso , Miguel de Carvalho , Ilaria Prosdocimi , Isadora Antoniano-Villalobos This is my paper

Pith reviewed 2026-05-18 15:25 UTC · model grok-4.3

classification 📊 stat.ME
keywords extreme value theorymixture modelsDirichlet processBayesian nonparametricgeneralized extreme valueblock maximaheterogeneous extremes
0
0 comments X

The pith

A Dirichlet process mixture of GEV distributions models heterogeneous block maxima while aligning with the extremal types theorem.

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

When extreme events fall into multiple hidden groups, the standard single Generalized Extreme Value distribution may not capture the data well. This paper shows that one can build alternative models for block maxima that still satisfy the theoretical limit law for extremes but offer more flexibility. The key step is to interpret the setup as a Bayesian nonparametric mixture with a Dirichlet process prior, allowing an infinite number of GEV components that adapt to the observed extremal behaviors. This avoids choosing the number of groups ahead of time and lets the model group similar blocks together based on their tail properties. The result is a practical tool illustrated on both artificial and real data sets.

Core claim

The paper establishes a mixture model for block maxima that has a Bayesian nonparametric interpretation as a Dirichlet process mixture of GEV distributions. By using an infinite number of components, it characterizes every possible block behavior and captures similarities between observations according to their extremal behavior, without needing to pre-specify the number of mixture components.

What carries the argument

Dirichlet process mixture of GEV distributions, which places a prior on the mixing measure to allow the data to determine the effective number of components for modeling heterogeneous extremes.

If this is right

  • The model remains consistent with the extremal types theorem.
  • It captures complex structures in extreme data without pre-specifying the number of components.
  • Similarities between observations are captured based on extremal behavior.
  • The approach applies directly to both simulated and real-world data.

Where Pith is reading between the lines

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

  • If the approach works, standard single-component GEV fits in many fields may need checking for latent heterogeneity.
  • Comparable nonparametric mixtures could be developed for other extreme value limit laws.
  • The model might be extended by adding covariates that help explain why blocks fall into different groups.

Load-bearing premise

That alternative block maxima-based models can be constructed to align with the extremal types theorem while allowing the mixture to represent heterogeneous groups effectively.

What would settle it

A simulation study where data is generated from a single GEV distribution and the fitted model is checked to see if it selects only one component with high posterior probability.

Figures

Figures reproduced from arXiv: 2509.15359 by Ilaria Prosdocimi, Isadora Antoniano-Villalobos, Miguel de Carvalho, Viviana Carcaiso.

Figure 1
Figure 1. Figure 1: Density of two-component block maxima based on Example 1. Fifty blocks are generated from two Exponential distributions, one with λ = 1 and the other with λ = 2; each block has size n = 1000. The red line represents the density that follows from (5). Example 1 (Two-component block maxima). Let F(x) = 1 − exp(−λx), for x > 0, be the cumulative distribution function of an exponential distribution with rate p… view at source ↗
Figure 2
Figure 2. Figure 2: Single-sample experiment. Left: posterior median density with 95% credible interval (shaded), compared to the true density (dashed line). Right: posterior median return level curve with credible interval (shaded), with true return levels (dashed line) and empirical quantiles (gray points). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Monte Carlo simulation results, II. Top: side-by-side violin plots of MISE. Bottom: side-by-side violin plots of 95% quantile fits from simulation experiments plotted against true 95% quantile. To assess the frequentist properties of the Bayesian procedure we use the mean integrated squared error (MISE), given by MISE = E[R R {f(z) − ˆf(z)} 2dz], where f is the true density under (9), and ˆf is the median … view at source ↗
Figure 3
Figure 3. Figure 3: Monte Carlo simulation results, I. Left: median posterior densities. Right: return levels. The dashed lines represent the corresponding true targets. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagnostics for model fit to seasonal precipitation maxima in Lisbon (1863–2018). Top left: median posterior density and 95% credible intervals (shaded) overlapping the histogram of the data, with a rug plot indicating individual observations colored by season. Top right: QQ plot of posterior median quantiles. Bottom left: posterior median return level curve with credible interval (shaded) and empirical qu… view at source ↗
Figure 6
Figure 6. Figure 6: Diagnostics for model fit to seasonal temperature maxima in Hong Kong (1884–2023). Top left: median posterior density and 95% credible intervals (shaded) overlapping the histogram of the data, with a rug plot indicating individual observations colored by season. Top right: QQ plot of posterior median quantiles. Bottom left: posterior median return level curve with credible interval (shaded) and empirical q… view at source ↗
read the original abstract

The conventional use of the Generalized Extreme Value (GEV) distribution to model block maxima may be inappropriate when extremes are actually structured into multiple heterogeneous groups. In this work, we propose a novel approach for describing the behavior of extreme values in the presence of such heterogeneity. Rather than defaulting to the GEV distribution simply because it arises as a theoretical limit, we show that alternative block maxima-based models can also align with the extremal types theorem while providing improved flexibility in practice. Our formulation leads us to a mixture model that has a Bayesian nonparametric interpretation as a Dirichlet process mixture of GEV distributions. The use of an infinite number of components enables the characterization of every possible block behavior, while at the same time capturing similarities between observations based on their extremal behavior. By employing a Dirichlet process prior on the mixing measure, we can capture the complex structure of the data without the need to pre-specify the number of mixture components. The application of the proposed model is illustrated using both simulated and real-world data.

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

1 major / 1 minor

Summary. The manuscript proposes a Dirichlet process mixture of GEV distributions as a model for block maxima under heterogeneity. It claims that this formulation aligns with the extremal types theorem while offering greater flexibility than a single GEV, interprets the model as a Bayesian nonparametric extension that captures all possible block behaviors via infinitely many components, and illustrates the approach on simulated and real data.

Significance. If the claimed alignment with the extremal types theorem can be rigorously established, the work would provide a flexible nonparametric Bayesian framework for modeling heterogeneous extremes without pre-specifying the number of components. The use of a Dirichlet process prior and the demonstration on both simulated and real data are positive features that could support broader adoption in extreme value applications.

major comments (1)
  1. [Abstract / theoretical development] Abstract and the theoretical development section: the central claim that the proposed mixture 'aligns with the extremal types theorem while providing improved flexibility' is load-bearing but unsupported by an explicit derivation. The Fisher-Tippett-Gnedenko theorem requires any non-degenerate limiting distribution of normalized block maxima to be a member of the GEV family; a non-degenerate mixture of GEVs is generally not itself GEV (e.g., it can exhibit multimodality or tail behavior outside any single GEV). No argument is given showing how heterogeneity in the underlying process produces a mixture as the limiting law rather than a single GEV.
minor comments (1)
  1. [Simulation study] The simulation study would benefit from an explicit statement of the data-generating process and the precise criteria used to assess recovery of heterogeneous structure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight the need for greater clarity in the theoretical justification. We address the major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: Abstract and the theoretical development section: the central claim that the proposed mixture 'aligns with the extremal types theorem while providing improved flexibility' is load-bearing but unsupported by an explicit derivation. The Fisher-Tippett-Gnedenko theorem requires any non-degenerate limiting distribution of normalized block maxima to be a member of the GEV family; a non-degenerate mixture of GEVs is generally not itself GEV (e.g., it can exhibit multimodality or tail behavior outside any single GEV). No argument is given showing how heterogeneity in the underlying process produces a mixture as the limiting law rather than a single GEV.

    Authors: We agree that the original manuscript lacks an explicit derivation and will revise the theoretical development section accordingly. When block maxima arise from heterogeneous processes (latent groups with distinct normalization constants or domains of attraction), the Fisher-Tippett-Gnedenko theorem applies conditionally on each group, yielding a GEV limit for that block. The unconditional distribution of observed maxima is then a mixture of GEVs. This formulation aligns with the theorem by extending it to heterogeneous settings, where a single GEV would be misspecified. The mixture's potential multimodality or varied tail behavior is a deliberate feature for capturing distinct extreme regimes rather than a violation. We will add this argument and clarify the distinction between conditional and unconditional limits. revision: yes

Circularity Check

0 steps flagged

No circularity: new modeling framework with independent content

full rationale

The paper introduces a Dirichlet process mixture of GEV distributions as a Bayesian nonparametric approach for heterogeneous block maxima. The central formulation is presented as a modeling choice that extends beyond the single GEV while claiming alignment with the extremal types theorem, without any derivation steps that reduce by the paper's own equations to quantities already fitted from the target data or to self-citations. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided derivation chain; the proposal remains self-contained as an alternative modeling framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the alignment of the mixture construction with the extremal types theorem and on the Dirichlet process prior enabling automatic component selection; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption Alternative block maxima-based models align with the extremal types theorem.
    Invoked to justify the mixture as a valid extension beyond the conventional GEV.

pith-pipeline@v0.9.0 · 5713 in / 1136 out tokens · 33352 ms · 2026-05-18T15:25:49.388095+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. How long should a block be?

    stat.ME 2026-05 unverdicted novelty 5.0

    Excessively long blocks lower asymptotic relative efficiency in the block-maxima method, and new likelihood and diagnostic procedures are proposed to check whether a chosen length is adequate under rounding or censoring.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · cited by 1 Pith paper

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" url if FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if ne...

    work page

  2. [2]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

    work page

  3. [3]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

    work page

  4. [4]

    write newline invisible empty 'skip invisible write newline if

    " write newline invisible empty 'skip invisible write newline if "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.checkb empty swap empty and 'skip 'new.block if FUNCTION field.or.null duplicate empty pop "" 'skip if FUNCTION emphasi...

    work page

  5. [5]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

    work page

  6. [6]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

    work page

  7. [7]

    write newline

    " write newline "" before.all 'output.state := FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if new.block note output fin.entry FUNCTION b...

    work page

  8. [8]

    Andr \'e , L. M. and Tawn, J. A. (2025), Gaussian mixture copulas for flexible dependence modelling in the body and tails of joint distributions, arXiv:2503.06255

    work page arXiv 2025

  9. [9]

    (2004), Statistics of Extremes : Theory and Applications , Hoboken, NJ: Wiley

    Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J. (2004), Statistics of Extremes : Theory and Applications , Hoboken, NJ: Wiley

    work page 2004

  10. [10]

    and Davison, A

    Boldi, M.-O. and Davison, A. C. (2007), A mixture model for multivariate extremes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69, 217--229

    work page 2007

  11. [11]

    and Castellanos, M

    Cabras, S. and Castellanos, M. E. (2011), A Bayesian approach for estimating extreme quantiles under a semiparametric mixture model, ASTIN Bulletin: The Journal of the IAA, 41, 87--106

    work page 2011

  12. [12]

    (2001), An Introduction to Statistical Modeling of Extreme Values , London: Springer

    Coles, S. (2001), An Introduction to Statistical Modeling of Extreme Values , London: Springer

    work page 2001

  13. [13]

    (2025), DATAstudio: The Research Data Warehouse of Miguel de Carvalho, R package version 1.2.1

    de Carvalho , M. (2025), DATAstudio: The Research Data Warehouse of Miguel de Carvalho, R package version 1.2.1

    work page 2025

  14. [14]

    de Carvalho , M., Huser, R., Naveau, P., and Reich, B. J. (2026), Handbook on Statistics of Extremes, Boca Raton, FL: Chapman & Hall/CRC

    work page 2026

  15. [15]

    and Ramirez, K

    de Carvalho , M. and Ramirez, K. P. (2025), Semiparametric Bayesian modeling of nonstationary joint extremes: how do big tech's extreme losses behave? Journal of the Royal Statistical Society, Ser. C, 74, 447--465

    work page 2025

  16. [16]

    Dunn, P. K. and Smyth, G. K. (1996), Randomized quantile residuals, Journal of Computational and Graphical Statistics, 5, 236--244

    work page 1996

  17. [17]

    Dunson, D. B. (2010), Nonparametric B ayes applications to biostatistics. In B ayesian N onparametrics, ( N . L . H jort et al., eds.), pp. 223--273, C ambridge U niversity P ress, C ambridge UK

    work page 2010

  18. [18]

    (2013), Modelling Extremal Events: For Insurance and Finance, New York: Springer

    Embrechts, P., Kl \"u ppelberg, C., and Mikosch, T. (2013), Modelling Extremal Events: For Insurance and Finance, New York: Springer

    work page 2013

  19. [19]

    Escobar, M. D. and West, M. (1995), Bayesian density estimation and inference using mixtures, Journal of the American Statistical Association, 90, 577--588

    work page 1995

  20. [20]

    Ferguson, T. S. (1973), Bayesian analysis of some nonparametric problems, The Annals of Statistics, 209--230

    work page 1973

  21. [21]

    --- (1974), Prior distributions on spaces of probability measures, The Annals of Statistics, 615--629

    work page 1974

  22. [22]

    (2011), Trends in frequency indices of daily precipitation over the I berian P eninsula during the last century, Journal of Geophysical Research: Atmospheres, 116

    Gallego, M., Trigo, R., Vaquero, J., Brunet, M., Garc \' a, J., Sigr \'o , J., and Valente, M. (2011), Trends in frequency indices of daily precipitation over the I berian P eninsula during the last century, Journal of Geophysical Research: Atmospheres, 116

    work page 2011

  23. [23]

    and Perron, F

    Guillotte, S. and Perron, F. (2016), Polynomial Pickands functions, Bernoulli, 22, 213--241

    work page 2016

  24. [24]

    (2011), Non-parametric Bayesian inference on bivariate extremes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 377--406

    Guillotte, S., Perron, F., and Segers, J. (2011), Non-parametric Bayesian inference on bivariate extremes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 377--406

    work page 2011

  25. [25]

    E., de Carvalho , M., and Chen, Y

    Hanson, T. E., de Carvalho , M., and Chen, Y. (2017), Bernstein polynomial angular densities of multivariate extreme value distributions, Statistics and Probability Letters, 128, 60--66

    work page 2017

  26. [26]

    and Huser, R

    Hazra, A. and Huser, R. (2021), Estimating high-resolution Red Sea surface temperature hotspots, using a low-rank semiparametric spatial model, The Annals of Applied Statistics, 15, 572--596

    work page 2021

  27. [27]

    In \'a cio de Carvalho , V., de Carvalho , M., and Branscum, A. J. (2017), Nonparametric Bayesian regression analysis of the Youden index, Biometrics, 73, 1279--1288

    work page 2017

  28. [28]

    and James, L

    Ishwaran, H. and James, L. F. (2001), Gibbs sampling methods for stick-breaking priors, Journal of the American Statistical Association, 96, 161--173

    work page 2001

  29. [29]

    --- (2002), Approximate Dirichlet process computing in finite normal mixtures : smoothing and prior information , Journal of Computational and Graphical Statistics, 11, 508--532

    work page 2002

  30. [30]

    E., and Walker, S

    Kalli, M., Griffin, J. E., and Walker, S. G. (2011), Slice sampling mixture models, Statistics and Computing, 21, 93--105

    work page 2011

  31. [31]

    C., and Tawn, J

    Lugrin, T., Davison, A. C., and Tawn, J. A. (2016), Bayesian uncertainty management in temporal dependence of extremes, Extremes, 19, 491--515

    work page 2016

  32. [32]

    (2011), A flexible extreme value mixture model, Computational Statistics & Data Analysis, 55, 2137--2157

    MacDonald, A., Scarrott, C., Lee, D., Darlow, B., Reale, M., and Russell, G. (2011), A flexible extreme value mixture model, Computational Statistics & Data Analysis, 55, 2137--2157

    work page 2011

  33. [33]

    A., and Antoniano-Villalobos, I

    Marcon, G., Padoan, S. A., and Antoniano-Villalobos, I. (2016), Bayesian inference for the extremal dependence, Electronic Journal of Statistics, 10, 3310--3337

    work page 2016

  34. [34]

    (2017), Mixture of extreme-value distributions: identifiability and estimation, Communications in Statistics---Theory and Methods, 46, 6528--6542

    Otiniano, C., Gon c alves, C., and Dorea, C. (2017), Mixture of extreme-value distributions: identifiability and estimation, Communications in Statistics---Theory and Methods, 46, 6528--6542

    work page 2017

  35. [35]

    E., Paiva, B

    Otiniano, C. E., Paiva, B. S., Vila, R., and Bourguignon, M. (2023), A bimodal model for extremes data, Environmental and Ecological Statistics, 30, 261--288

    work page 2023

  36. [36]

    A., M \"u ller, P., Jara, A., and MacEachern, S

    Quintana, F. A., M \"u ller, P., Jara, A., and MacEachern, S. N. (2022), The dependent D irichlet process and related models, Statistical Science, 37, 24--41

    work page 2022

  37. [37]

    P., de Carvalho , M., and Gutierrez, L

    Ramirez, V. P., de Carvalho , M., and Gutierrez, L. (2025, to appear), Heavy-tailed NGG -mixture models, Bayesian Analysis

    work page 2025

  38. [38]

    Reich, B. J. and Ghosh, S. K. (2019), Bayesian Statistical Methods, Boca Raton, FL: Chapman and Hall/CRC

    work page 2019

  39. [39]

    A., and Brown, S

    Richards, J., Tawn, J. A., and Brown, S. (2023), Joint estimation of extreme spatially aggregated precipitation at different scales through mixture modelling, Spatial Statistics, 53, 100725

    work page 2023

  40. [40]

    and Kafadar, K

    Rodu, J. and Kafadar, K. (2022), The q--q Boxplot, Journal of Computational and Graphical Statistics, 31, 26--39

    work page 2022

  41. [41]

    (1994), A constructive definition of Dirichlet priors, Statistica Sinica, 639--650

    Sethuraman, J. (1994), A constructive definition of Dirichlet priors, Statistica Sinica, 639--650

    work page 1994

  42. [42]

    Stephenson, A. G. (2002), evd : Extreme value distributions, R News, 2, 31--32

    work page 2002

  43. [43]

    (1910), Introduction \`a la Th \'e orie des Fonctions d'une Variable , vol

    Tannery, J. (1910), Introduction \`a la Th \'e orie des Fonctions d'une Variable , vol. 1, A. Hermann

    work page 1910

  44. [44]

    (2023), Modeling the extremes of bivariate mixture distributions with application to oceanographic data, Journal of the American Statistical Association, 118, 1373--1384

    Tendijck, S., Eastoe, E., Tawn, J., Randell, D., and Jonathan, P. (2023), Modeling the extremes of bivariate mixture distributions with application to oceanographic data, Journal of the American Statistical Association, 118, 1373--1384

    work page 2023

  45. [45]

    A., Trigo, R., Barros, M., Nunes, L

    Valente, M. A., Trigo, R., Barros, M., Nunes, L. F., Alves, E., Pinhal, E., Coelho, F., Mendes, M., and Miranda, J. (2008), Early stages of the recovery of Portuguese historical meteorological data, in MEDARE-Proceedings of the International Workshop on Rescue and Digitization of Climate Records in the Mediterranean Basin, WCDMP, no. 67, pp. 95--102

    work page 2008