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REVIEW 2 major objections 4 minor 76 references

A zero-inflated mixed-effects spatial point process for grouped storm loss data

T0 review · 2 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read A latent zero-inflated point process of claims lets insurers keep fine-scale weather and property predictors when losses are only counted by county or community.

desk verdict Solid, usable extension of Gao–Shi to latent point processes for unbalanced grouped storm counts; predictive gains are real but partly from property-level exposure detail rather than the derivation alone. read the letter →

arxiv 2607.03852 v1 pith:MQZYQOCD submitted 2026-07-04 stat.AP

Lisa Gao , Sébastien Jessup , Tianxing Yan This is my paper · ORCID
classification stat.AP
keywords spatialpointprocesszero-inflationmisalignmentmultivariatecountmodelingstormlosspredictionEMalgorithminsuranceclaims
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Insurers increasingly have dense weather and exposure maps, but claim outcomes often arrive only as counts by county or similar regions, so standard practice averages the predictors and loses local signal. This paper treats unobserved claim locations inside each storm as a zero-inflated mixed Poisson point process and derives the exact multivariate zero-inflated negative-binomial distribution of the observed regional counts. The construction keeps the fine-scale predictors inside the intensity integrals, captures joint excess zeros and within-storm dependence through a shared gamma random effect, and supplies a closed-form likelihood estimated by a tailored EM algorithm. On Texas flood and flash-flood episodes linked to NFIP policies, models that retain the granular predictors fit better and forecast both point and joint distributions more accurately than the same multivariate zero-inflated count model run on predictors aggregated to the same regions, with the largest gains when the observed losses are coarsest.

What carries the argument

The zero-inflated mixed-effects spatial point process: with probability p_i the storm produces the empty pattern; otherwise claims form an inhomogeneous Poisson process whose intensity is a storm-specific gamma random effect times a baseline times exp(x(s)´β). The induced multivariate count law on any partition is closed-form and nests ordinary multivariate ZINB regression when predictors are already aggregated.

What would settle it

Re-estimate the same models after replacing the constructed NOAA–NFIP linkages with ground-truth geocoded claim locations (or an independent high-quality storm-to-claim matching) and check whether the predictive gains of granular predictors over aggregated predictors disappear or reverse.

Watch

Extended reading notes

Core claim

The joint distribution of claim counts across arbitrary disjoint subregions of a storm is a multivariate zero-inflated negative binomial whose mean parameters are spatial integrals of a log-linear intensity driven by granular predictors; that distribution is obtained exactly from a latent zero-inflated mixed Poisson point process, so fine-scale weather and exposure information can be retained even when only grouped counts are observed.

Load-bearing premise

The constructed storm windows, date matching, and flood/flash-flood event definitions must faithfully link the public storm and NFIP data so that estimated local intensity effects reflect real storm heterogeneity rather than matching artifacts.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper models spatially grouped storm claim counts by assuming a latent zero-inflated mixed Poisson spatial point process of unobserved claim locations, then deriving the induced multivariate zero-inflated negative binomial distribution for arbitrary regional partitions (Eq. 3). Granular weather and exposure predictors enter the intensity via spatial integrals approximated by Berman–Turner quadrature; storm-level zero-inflation and a gamma random effect induce within-storm dependence. An EM algorithm is developed for the closed-form likelihood. Simulations recover parameters with low bias and near-nominal coverage across aggregation levels; an application to NOAA flood/flash-flood episodes linked to NFIP exposures and claims in Texas reports better in-sample fit and out-of-sample point and probabilistic scores than multivariate ZINB regression that uses only region-aggregated covariates.

Significance. If the results hold, the paper supplies a coherent, likelihood-based route for insurers to use densely measured weather and exposure data when claim outcomes are only available as unbalanced areal counts. Strengths include a clean derivation of the multivariate ZINB from the latent process, a tractable EM procedure, simulation evidence of √m-consistency and coverage across aggregation levels (Table 1), and a real-data comparison with proper scoring rules and Diebold–Mariano tests (Tables 5–8). The framework is agnostic to region definitions and nests the Type II MZINB of Zhang et al. (2025) when predictors are also aggregated, which is a useful contribution to actuarial multivariate count modeling and spatial misalignment for discrete losses.

major comments (2)
  1. §6 and Table A1 vs. §3/§6 quadrature: the central claim that the point-process derivation improves fit and prediction over MZINB (Tables 5–8) is not cleanly isolated from the granularity of the exposure representation. The proposed model evaluates intensity at individual NFIP property locations (Berman–Turner points = exposures), so µ(Bij) is effectively a sum of property-level intensities; the MZINB uses only region averages/proportions. Gains can therefore arise from retaining property-level detail rather than from the latent continuous intensity or spatial-integral construction per se. A load-bearing control is needed: e.g., an MZINB (or other count model) that uses the same property-level exposure features aggregated only as needed for the outcome regions, or a non-point-process property-level intensity model with the same zero-inflation/mixing structure. Without that, the methodolog
  2. §5 (data construction and limitations): the NOAA episode–NFIP linkage (convex-hull windows, one-week date margin, jittered 0.1° locations, flood/flash-flood as pluvial proxies) is acknowledged as imperfect, but estimated intensity coefficients and predictive gains are still interpreted as reflecting localized storm heterogeneity. Matching artifacts (misattributed claims, window over/under-coverage, fluvial vs. pluvial mix) could inflate apparent value of granular predictors. Sensitivity checks—alternative windows, stricter date matching, or restricting to high-confidence episodes—would strengthen the application claim that is used to sell the method.
minor comments (4)
  1. Table 1: coverage for some parameters (e.g., β1 at m=1000, 5×5) is slightly below 95%; a brief note on whether this is Monte Carlo noise or undercoverage of the observed Hessian would help.
  2. §2.1–2.2: the term “multivariate negative binomial” is used for the negative multinomial component; a short clarification that margins are negative binomial and the joint is negative multinomial would reduce ambiguity relative to other actuarial multivariate NB constructions.
  3. Figure 9 and prediction section: intensity is shown at exposure locations for one storm; a brief statement on how intensity is evaluated at non-exposure locations (if needed for maps) would aid reproducibility.
  4. Appendix Algorithm A1 and Table A1 are useful; ensure cross-references in the main text point to them consistently.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity: multivariate ZINB is derived from a latent point process, nested MZINB is the aggregated special case, and gains are tested on held-out storms.

  1. self citation load bearing [§2.1–2.2, Eq. (3); citation to Gao and Shi (2022)]
    "The mixed Poisson point process component builds on Gao and Shi (2022)... The above result follows from Property 2 of the mixed-effects Poisson point process in Gao and Shi (2022), where the specific form of the multivariate negative binomial component in (3) is the negative multinomial distribution"

    One co-author’s prior paper is cited for the mixed Poisson → multivariate NB property used in Eq. (3). This is not load-bearing circularity: the property is classical probability (gamma mixture of independent Poissons yields negative multinomial margins), the present paper adds zero-inflation, spatial misalignment, and EM estimation, and the empirical claim is tested against held-out storms and a nested aggregated-predictor baseline. Flagged only as minor self-citation, not as forcing the result.

full rationale

The paper’s central chain is a standard stochastic-process derivation, not a fit renamed as prediction. A zero-inflated mixed Poisson point process (Eqs. 1–2) induces, for arbitrary disjoint regions, the multivariate zero-inflated negative binomial pmf (Eq. 3) via the classical gamma-mixed Poisson / negative-multinomial construction; parameters are estimated by EM/MLE on 2009–2019 storms and scored on 2020–2023 storms (Tables 5–8). The MZINB regression comparator is explicitly the nested special case when predictors are aggregated to the same regions as the counts, so outperformance is an empirical comparison of granular vs aggregated covariates under a shared dependence structure, not a tautology. The only self-citation of note is Gao and Shi (2022) for the mixed Poisson point-process component and its Property 2; that is ordinary cumulative research on a standard mixture result, not a uniqueness theorem or ansatz that forces the present claims. Skeptical concerns about incomplete isolation of the derivation from property-level quadrature (Berman–Turner at NFIP locations) are identification/design issues, not circularity by construction. Score 1 only for that minor non-load-bearing self-citation; no step reduces the target result to its inputs by definition.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central empirical claim rests on a standard spatial point-process generative model (Poisson intensity with gamma mixing and storm-level zero inflation), numerical quadrature, and a carefully constructed but imperfect public storm–exposure–claim dataset. Free parameters are the usual regression and dispersion coefficients. No new physical entities are postulated; the ‘latent claim point process’ is a modeling device, not an independently measured object.

free parameters (4)
  • zero-inflation coefficients η
    Logit coefficients for storm-level excess-zero probability; fitted by EM/MLE on storm indicators and storm-level covariates.
  • intensity coefficients β and baseline γ
    Log-linear intensity parameters for granular predictors and common baseline rate; fitted via spatial integrals in the multivariate ZINB likelihood.
  • gamma mixing scale ψ
    Storm-level unobserved heterogeneity / overdispersion parameter; fitted by MLE/EM (initialized by moment matching).
  • quadrature grid density (e.g. 50×50 in simulations)
    Numerical choice controlling approximation of μ_i(B_ij); not estimated from data but affects likelihood evaluation.
assumptions (6)
  • domain assumption Conditional on storm random effect U_i, claim locations form an inhomogeneous Poisson point process with intensity U_i γ exp{x_i(s)^T β}.
    §2.1; standard mixed Poisson point process assumption transferred from Gao and Shi (2022).
  • domain assumption Storm-specific random effects U_i are i.i.d. gamma with mean 1 and variance ψ.
    §2.1; chosen for closed-form negative multinomial margins and tractability.
  • domain assumption Excess zeros arise from a storm-level Bernoulli/logit mixture independent of the spatial pattern given the zero-inflation component.
    Eq. (1); standard zero-inflation mixture at the storm (not region) level.
  • domain assumption Storms are independent replicates; within-storm dependence is only through shared p_i and U_i.
    §2–3 likelihood factorization as sum of storm contributions.
  • ad hoc to paper Berman–Turner quadrature with exposure locations adequately approximates spatial intensity integrals over counties/communities.
    §3 Eq. (10); practical approximation tied to available exposure points rather than a dense regular lattice everywhere.
  • ad hoc to paper NOAA flood/flash-flood episodes linked to NFIP claims via convex-hull windows and date proximity represent the relevant pluvial storm loss process.
    §5 data construction; authors note non-exclusive causes and coverage mismatches.
invented entities (1)
  • Latent zero-inflated mixed-effects spatial point process of unobserved claim locations for grouped outcomes
    purpose: Bridge granular predictors to areal multivariate counts and induce within-storm dependence in zeros and positive counts.
    Modeling construct; not independently observed in this paper (unlike Gao and Shi 2022, where points were observed). Falsifiable only indirectly via predictive scores on grouped counts.

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Cite this review

Pith. "Pith review of A zero-inflated mixed-effects spatial point process for grouped storm loss data." pith.science (2026). https://pith.science/paper/MQZYQOCD

@misc{pith2026260703852,
  author       = {Pith},
  title        = {Pith review of: A zero-inflated mixed-effects spatial point process for grouped storm loss data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQZYQOCD}},
  note         = {Machine review of arXiv:2607.03852}
}
read the original abstract

The increasing granularity of third-party weather and exposure information can allow insurers to more effectively predict weather-related losses. However, loss outcomes are often reported in spatially grouped observations, such as at the county level, so higher resolution predictors are aggregated to align with the granularity of the outcome in standard analyses. Assuming an underlying zero-inflated mixed-effects spatial point process framework for claims arising from a common storm, we derive a model for unbalanced, multivariate zero-inflated count data that incorporates rich weather and exposure predictors observed at higher spatial granularity to predict claim patterns. The model accommodates the dependence between locations affected by a common storm in the excess zeros, as well as in the joint claim counts. Using real property exposure and loss data, we emphasize the value of incorporating granular predictors to address the localized heterogeneity of storm losses.

Figures

Figures reproduced from arXiv: 2607.03852 by the authors.

Figure 1
Figure 1. Distribution of storm-level claim frequency for m = 1,000 storms from one simulation replication. In particular, we consider different extents of spatial aggregation of the grouped loss out￾comes, and divide W into b × b grid-based subregions, for b ∈ {2, 3, 4, 5}. As an example, [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Example of an underlying simulated spatial point pattern, with observed claim counts spatially grouped into 3 × 3 subregions [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Boxplots of the number of EM algorithm iterations until convergence under various levels of spatial aggregation of the loss outcomes, using 500 storms (left) and 1,000 storms (right), based on 200 simulation replications. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Storm windows for 2023 storms in Texas as blue polygons, with exposure locations marked in red. tively produced 7,337 claims. For each storm, we observe characteristics of the affected properties and associated claims, but FEMA does not provide an identifier that allow…
Figure 5
Figure 5. Figure 5: Example storm map with spatial window outlined in blue, exposure locations as black points, and counties color-coded by the number of observed claims. the ceiling). Similarly, the flood and flash flood storm events may lead to water property damage, yet are not exhaust…
Figure 6
Figure 6. Figure 6: Distribution of storm-level claim frequency. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: Q-Q plots of the non-randomized PIT of storm-level claim counts for the ZI point process (left) and MZINB regression (right), at the county (top) and community (bottom) levels of grouped outcomes [PITH_FULL_IMAGE:figures/full_fig_p034_7.png]
Figure 8
Figure 8. Figure 8: Reliability plots comparing the storm-level fitted probabilities of zero claims and the observed proportion of storms with zero claims, for the ZI point process (left) and MZINB regres￾sion (right), at the county (top) and community (bottom) levels of grouped outcomes.…
Figure 9
Figure 9. Figure 9: Predicted zero-inflated spatial point process claim intensity at exposure locations for an out-of-sample storm, with observed community-level claim frequencies indicated by colored squares. et al., 2009) [PITH_FULL_IMAGE:figures/full_fig_p036_9.png]

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Reference graph

Works this paper leans on

76 extracted references · 1 canonical work pages

  1. [1]

    Insurance: Mathematics and Economics , volume=

    Leveraging high-resolution weather information to predict hail damage claims: A spatial point process for replicated point patterns , author=. Insurance: Mathematics and Economics , volume=. 2022 , publisher=

    2022

  2. [2]

    Biometrics , volume=

    Zero-inflated Poisson and binomial regression with random effects: a case study , author=. Biometrics , volume=. 2000 , publisher=

    2000

  3. [3]

    Journal of atmospheric and oceanic technology , volume=

    An overview of the global historical climatology network-daily database , author=. Journal of atmospheric and oceanic technology , volume=

  4. [4]

    1993 , publisher=

    Statistics for spatial data , author=. 1993 , publisher=

    1993

  5. [5]

    Journal of the royal statistical society: series B (methodological) , volume=

    Maximum likelihood from incomplete data via the EM algorithm , author=. Journal of the royal statistical society: series B (methodological) , volume=. 1977 , publisher=

    1977

  6. [6]

    2008 , publisher=

    The EM algorithm and extensions , author=. 2008 , publisher=

    2008

  7. [7]

    2002 , publisher=

    Statistical inference , author=. 2002 , publisher=

    2002

  8. [8]

    Biometrics , volume=

    Predictive model assessment for count data , author=. Biometrics , volume=. 2009 , publisher=

    2009

Show all 76 references
  1. [9]

    Elevated Building , year =

  2. [10]

    Mathematical Geology , volume=

    An experimental comparison of ordinary and universal kriging and inverse distance weighting , author=. Mathematical Geology , volume=. 1999 , publisher=

    1999

  3. [11]

    arXiv preprint arXiv:1611.00519 , year=

    On the convergence of the EM algorithm: A data-adaptive analysis , author=. arXiv preprint arXiv:1611.00519 , year=

    arXiv

  4. [12]

    2023 , howpublished =

    2023

  5. [13]

    Encyclopedia of Mathematical Geosciences , pages=

    Inverse distance weight , author=. Encyclopedia of Mathematical Geosciences , pages=. 2021 , publisher=

    2021

  6. [14]

    Water Damage Statistics: Key Insights and Trends for Homeowners , year =

  7. [15]

    Between 179.8 and 496.0 Billion Each Year , year=

    Flooding Costs the U.S. Between 179.8 and 496.0 Billion Each Year , year=

  8. [16]

    Impacts, Risks, and Adaptation in the United States: Fourth National Climate Assessment, Volume II , editor =

    Upmanu Lall and Thomas Johnson and Patrick Colohan and Amir Aghakouchak and Casey Brown and Gregory McCabe and Roger Pulwarty and Ashok Sankarasubramanian , title =. Impacts, Risks, and Adaptation in the United States: Fourth National Climate Assessment, Volume II , editor =. ...

    2018

  9. [17]

    2003 , publisher=

    Statistical inference and simulation for spatial point processes , author=. 2003 , publisher=

    2003

  10. [18]

    ASTIN Bulletin , author=

    A representation-learning approach for insurance pricing with images , volume=. ASTIN Bulletin , author=. 2024 , pages=. doi:10.1017/asb.2024.9 , number=

    2024 doi

  11. [19]

    Computo , year =

    Delattre, Maud and Kuhn, Estelle , publisher =. Computo , year =. doi:10.57750/r5gx-jk62 , issn =

    doi

  12. [20]

    Louis , title =

    Thomas A. Louis , title =. Journal of the Royal Statistical Society: Series B (Methodological) , year =

  13. [21]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume=

    Assessing hail risk for property insurers with a dependent marked point process , author=. Journal of the Royal Statistical Society Series A: Statistics in Society , volume=. 2022 , publisher=

    2022

  14. [22]

    Wiley Interdisciplinary Reviews: Computational Statistics , volume=

    Insurance risk assessment in the face of climate change: Integrating data science and statistics , author=. Wiley Interdisciplinary Reviews: Computational Statistics , volume=. 2019 , publisher=

    2019

  15. [23]

    Finding and Using Climate Data , year =

  16. [24]

    2025 , url =

    Bowen, Steve and Kerschner, Brian and Ng, Jin Zheng , title =. 2025 , url =

    2025

  17. [25]

    2023 , url =

    Severe Convective Storms: Decoding the Dynamics , institution =. 2023 , url =

    2023

  18. [26]

    2025 , url =

    Climate and Catastrophe Insight , institution =. 2025 , url =

    2025

  19. [27]

    2020 , url =

    Severe Convective Storms: Evolving risks call for innovation to reduce costs, drive resilience , institution =. 2020 , url =

    2020

  20. [28]

    Maximize actionable underwriting data at bind and renewal , year =

  21. [29]

    Spatial Hazard Events and Losses Database for the United States , year =

  22. [30]

    Catalogue , year =

    Catastrophe Indices and Quantification Inc. Catalogue , year =

  23. [31]

    2015 , publisher=

    Statistics for spatial data , author=. 2015 , publisher=

    2015

  24. [32]

    Journal of the American Statistical Association , volume=

    Combining incompatible spatial data , author=. Journal of the American Statistical Association , volume=. 2002 , publisher=

    2002

  25. [33]

    Handbook of spatial statistics , volume=

    Misaligned Spatial Data; The Change of Support Problem , author=. Handbook of spatial statistics , volume=. 2010 , publisher=

    2010

  26. [34]

    Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=

    Approximating point process likelihoods with GLIM , author=. Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=. 1992 , publisher=

    1992

  27. [35]

    Scandinavian Actuarial Journal , volume=

    Spatial modelling of claim frequency and claim size in non-life insurance , author=. Scandinavian Actuarial Journal , volume=. 2007 , publisher=

    2007

  28. [36]

    Scandinavian Actuarial Journal , volume=

    Spatial statistical modelling of insurance risk: a spatial epidemiological approach to car insurance , author=. Scandinavian Actuarial Journal , volume=. 2019 , publisher=

    2019

  29. [37]

    Scandinavian Actuarial Journal , volume=

    Spatial modelling of risk premiums for water damage insurance , author=. Scandinavian Actuarial Journal , volume=. 2022 , publisher=

    2022

  30. [38]

    Insurance: Mathematics and Economics , volume=

    Multivariate negative binomial models for insurance claim counts , author=. Insurance: Mathematics and Economics , volume=. 2014 , publisher=

    2014

  31. [39]

    ASTIN Bulletin: The Journal of the IAA , volume=

    A new multivariate zero-inflated hurdle model with applications in automobile insurance , author=. ASTIN Bulletin: The Journal of the IAA , volume=. 2022 , publisher=

    2022

  32. [40]

    2008 , publisher=

    Stochastic claims reserving methods in insurance , author=. 2008 , publisher=

    2008

  33. [41]

    2012 , publisher=

    Loss reserving: an actuarial perspective , author=. 2012 , publisher=

    2012

  34. [42]

    North American Actuarial Journal , volume=

    Storm cat bond: Modeling and valuation , author=. North American Actuarial Journal , volume=. 2024 , publisher=

    2024

  35. [43]

    Scandinavian Actuarial Journal , volume=

    Future building water loss projections posed by climate change , author=. Scandinavian Actuarial Journal , volume=. 2011 , publisher=

    2011

  36. [44]

    North American Actuarial Journal , volume=

    Factor copula approaches for assessing spatially dependent high-dimensional risks , author=. North American Actuarial Journal , volume=. 2017 , publisher=

    2017

  37. [45]

    ASTIN Bulletin: The Journal of the IAA , volume=

    Risk modeling of property insurance claims from weather events , author=. ASTIN Bulletin: The Journal of the IAA , volume=. 2025 , publisher=

    2025

  38. [46]

    Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=

    A Bayesian hierarchical model with spatial variable selection: the effect of weather on insurance claims , author=. Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=. 2013 , publisher=

    2013

  39. [47]

    Hydrology and Earth System Sciences , volume=

    A statistical analysis of insurance damage claims related to rainfall extremes , author=. Hydrology and Earth System Sciences , volume=. 2013 , publisher=

    2013

  40. [48]

    Risks , volume=

    The Italian Actuarial Climate Index: A National Implementation Within the Emerging European Framework , author=. Risks , volume=. 2025 , publisher=

    2025

  41. [49]

    Risks , volume=

    Multivariate frequency-severity regression models in insurance , author=. Risks , volume=. 2016 , publisher=

    2016

  42. [50]

    Insurance: Mathematics and Economics , volume=

    Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution , author=. Insurance: Mathematics and Economics , volume=. 2019 , publisher=

    2019

  43. [51]

    ASTIN Bulletin: The Journal of the IAA , volume=

    A class of mixture of experts models for general insurance: Application to correlated claim frequencies , author=. ASTIN Bulletin: The Journal of the IAA , volume=. 2019 , publisher=

    2019

  44. [52]

    2025 , url =

    Hornack, Julia and Schmidt, Jeffrey and Filimonov, Vadim , title =. 2025 , url =

    2025

  45. [53]

    2021 , url =

    Natural Catastrophes in 2020: Secondary Perils in Spotlight , institution =. 2021 , url =

    2020

  46. [54]

    Insurance: Mathematics and Economics , volume=

    The multivariate mixed negative binomial regression model with an application to insurance a posteriori ratemaking , author=. Insurance: Mathematics and Economics , volume=. 2021 , publisher=

    2021

  47. [55]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume=

    Multivariate claim count regression model with varying dispersion and dependence parameters , author=. Journal of the Royal Statistical Society Series A: Statistics in Society , volume=. 2023 , publisher=

    2023

  48. [56]

    Insurance: Mathematics and Economics , volume=

    Sarmanov family of multivariate distributions for bivariate dynamic claim counts model , author=. Insurance: Mathematics and Economics , volume=. 2016 , publisher=

    2016

  49. [57]

    Journal of the Royal Statistical Society Series C: Applied Statistics , volume=

    Investigating the effect of climate-related hazards on claim frequency prediction in motor insurance with incomplete data , author=. Journal of the Royal Statistical Society Series C: Applied Statistics , volume=. 2025 , publisher=

    2025

  50. [58]

    Storm Events Database , year = 2025, howpublished =

    2025

  51. [59]

    Storm Data Preparation , year = 2021, howpublished =

    2021

  52. [60]

    Wharton Risk Management and Decision Processes Center , year=

    The emerging private residential flood insurance market in the United States , author=. Wharton Risk Management and Decision Processes Center , year=

  53. [61]

    Journal of economic perspectives , volume=

    Catastrophe economics: the national flood insurance program , author=. Journal of economic perspectives , volume=. 2010 , publisher=

    2010

  54. [62]

    National Flood Insurance Program Claims Handbook , year = 2024, howpublished =

    2024

  55. [63]

    OpenFEMA Dataset: FIMA NFIP Redacted Policies - v2 , year = 2025, howpublished =

    2025

  56. [64]

    OpenFEMA Dataset: FIMA NFIP Redacted Claims - v2 , year = 2025, howpublished =

    2025

  57. [65]

    International edition, New Jersey: Prentice Hall , pages=

    Econometric analysis 4th edition , author=. International edition, New Jersey: Prentice Hall , pages=

  58. [66]

    Annals of economic and social measurement , volume=

    Estimation and inference in nonlinear structural models , author=. Annals of economic and social measurement , volume=

  59. [67]

    Journal of the Royal Statistical Society: Series B (Methodological) , volume=

    A general definition of residuals , author=. Journal of the Royal Statistical Society: Series B (Methodological) , volume=. 1968 , publisher=

    1968

  60. [68]

    Journal of the Royal Statistical Society: Series D (The Statistician) , volume=

    The comparison and evaluation of forecasters , author=. Journal of the Royal Statistical Society: Series D (The Statistician) , volume=. 1983 , publisher=

    1983

  61. [69]

    Journal of the American statistical Association , volume=

    Strictly proper scoring rules, prediction, and estimation , author=. Journal of the American statistical Association , volume=. 2007 , publisher=

    2007

  62. [70]

    Monthly Weather Review , volume=

    Variogram-based proper scoring rules for probabilistic forecasts of multivariate quantities , author=. Monthly Weather Review , volume=

  63. [71]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Probabilistic forecasts, calibration and sharpness , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2007 , publisher=

    2007

  64. [72]

    Australian & New Zealand Journal of Statistics , volume=

    Practical Maximum Pseudolikelihood for Spatial Point Patterns: (with Discussion) , author=. Australian & New Zealand Journal of Statistics , volume=. 2000 , publisher=

    2000

  65. [73]

    Energy and AI , volume=

    An introduction to multivariate probabilistic forecast evaluation , author=. Energy and AI , volume=. 2021 , publisher=

    2021

  66. [74]

    Communications in Statistics-Theory and Methods , volume=

    A comparative analysis of several multivariate zero-inflated and zero-modified models with applications in insurance , author=. Communications in Statistics-Theory and Methods , volume=. 2025 , publisher=

    2025

  67. [75]

    1979 , publisher=

    Theoretical statistics , author=. 1979 , publisher=

    1979

  68. [76]

    2016 , publisher=

    Spatial point patterns: methodology and applications with R , author=. 2016 , publisher=

    2016

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