arXiv subjects

Hélène Cossette

Publications and source records attributed to Hélène Cossette.

At least 19 recordsLinked to original sources

arXiv2026

Tree-structured Ising models under mean parameterization

In the risk modeling literature, the Ising model has emerged as a valuable framework for dependent Bernoulli random variables, as its underlying graphical structure captures complex dependence patterns. It is almost always employed in its exponential-family parameterization, which comes with some drawbacks such as the marginal distributions not being fixed and an intractable normalizing constant. We study tree-structured Ising models under their mean parameterization, particularly well suited to risk modeling applications, especially as it eliminates these drawbacks. We study tree-structured Ising models under their mean parameterization, which eliminates these drawbacks and is particularly attractive for risk modeling. The mean parameterization also facilitates the characterization of dependence structures and the distribution of aggregate risks. In particular, we derive an analytic expression for the joint probability generating function, yielding an efficient method for computing the aggregate-risk distribution. Similarly, we obtain the ordinary generating function of expected allocations allowing exact computations for risk allocation. The mean parameterization also allows for a stochastic representation of Ising models, yielding a direct sampling algorithm otherwise unavailable under the exponential-family parameterization. We show that Markov random fields with fixed Poisson marginal distributions provide an efficient and accurate approximation for tree-structured Ising models, in the spirit of Poisson approximations.

math.ST↗

arXiv2026

On a risk model with tree-structured Poisson Markov random field frequency, with application to rainfall events

In many insurance contexts, dependence between risks of a portfolio may arise from their frequencies. We investigate a dependent risk model in which we assume the vector of count variables to be a tree-structured Markov random field with Poisson marginals. The tree structure translates into a wide variety of dependence schemes. We study the global risk of the portfolio and the risk allocation to all its constituents. We provide asymptotic results for portfolios defined on infinitely growing trees. To illustrate its flexibility and computational scalability to higher dimensions, we calibrate the risk model on real-world extreme rainfall data and perform a risk analysis.

stat.ME↗

arXiv2026

Conway--Maxwell multivariate Bernoulli distribution

We investigate the Conway--Maxwell multivariate Bernoulli distributions, a family of multivariate Bernoulli distributions derived from the Conway--Maxwell-binomial distribution. We show that it is possible to set the parametrization such that the Bernoulli marginals remain intact, allowing us to study dependence properties within this family. In particular, we demonstrate that this family spans the full spectrum of dependence. Moreover, for specific ranges of the parameters, these distributions satisfy the strongly Rayleigh property, a negative dependence notion stronger than negative association.

math.ST↗

arXiv2025

Centrality and shape-related comparisons in a tree-structured Markov random field

Understanding the effects of the choice of the tree on the joint distribution of a tree-structured Markov random field (MRF) is crucial for fully exploiting the intelligibility of such probabilistic graphical models. Tools must be developed in this regard: this is the overarching objective of this paper. Our discussion is two-fold. First, we examine concepts specific to network centrality theory. We put forth a new conception of centrality for MRFs that not only accounts for the tree topology, but also the underlying stochastic dynamics. In this vein, we compare synecdochic pairs, random vectors comprising a MRF's component and its sum, using stochastic orders. The resulting orderings are transferred to risk-allocation quantities, which therefore serve as new centrality indices tailored to our stochastic framework. Second, we shed light on the influence of the tree's shapes, by establishing convex orderings for MRFs encrypted on trees of different shapes. This results in the design of a new partial order on tree shapes. This analysis is done within the framework of a propagation-based family of tree-structured MRFs with the uncommon property of having fixed Poisson marginal distributions unaffected by the dependence scheme. This work is a first step into the analysis of MRFs' trees' shapes and a stepping stone to extending this analysis to a broader framework.

math.ST↗

arXiv2025

Extremal negative dependence and the strongly Rayleigh property

We provide a geometrical characterization of extremal negative dependence as a convex polytope in the simplex of multidimensional Bernoulli distributions, and we prove that it is an antichain that satisfies some minimality conditions with respect to the strongest negative dependence orders. We study the strongly Rayleigh property within this class and explicitly find a distribution that satisfies this property by maximizing the entropy. Furthermore, we construct a chain for the supermodular order starting from extremal negative dependence to independence by mixing the maximum entropy strongly Rayleigh distribution with independence.

math.PR↗

arXiv2025

Efficient evaluation of risk allocations

Expectations of marginals conditional on the total risk of a portfolio are crucial in risk-sharing and allocation. However, computing these conditional expectations may be challenging, especially in critical cases where the marginal risks have compound distributions or when the risks are dependent. We introduce a generating function method to compute these conditional expectations. We provide efficient algorithms to compute the conditional expectations of marginals given the total risk for a portfolio of risks with lattice-type support. We show that the ordinary generating function of unconditional expected allocations is a function of the multivariate probability generating function of the portfolio. The generating function method allows us to develop recursive and transform-based techniques to compute the unconditional expected allocations. We illustrate our method to large-scale risk-sharing and risk allocation problems, including cases where the marginal risks have compound distributions, where the portfolio is composed of dependent risks, and where the risks have heavy tails, leading in some cases to computational gains of several orders of magnitude. Our approach is useful for risk-sharing in peer-to-peer insurance and risk allocation based on Euler's rule.

stat.AP↗

arXiv2025

Tree-structured Markov random fields with Poisson marginal distributions

A new family of tree-structured Markov random fields for a vector of discrete counting random variables is introduced. According to the characteristics of the family, the marginal distributions of the Markov random fields are all Poisson with the same mean, and are untied from the strength or structure of their built-in dependence. This key feature is uncommon for Markov random fields and most convenient for applications purposes. The specific properties of this new family confer a straightforward sampling procedure and analytic expressions for the joint probability mass function and the joint probability generating function of the vector of counting random variables, thus granting computational methods that scale well to vectors of high dimension. We study the distribution of the sum of random variables constituting a Markov random field from the proposed family, analyze a random variable's individual contribution to that sum through expected allocations, and establish stochastic orderings to assess a wide understanding of their behavior.

stat.ME↗

arXiv2024

Generalized FGM dependence: Geometrical representation and convex bounds on sums

Building on the one-to-one relationship between generalized FGM copulas and multivariate Bernoulli distributions, we prove that the class of multivariate distributions with generalized FGM copulas is a convex polytope. Therefore, we find sharp bounds in this class for many aggregate risk measures, such as value-at-risk, expected shortfall, and entropic risk measure, by enumerating their values on the extremal points of the convex polytope. This is infeasible in high dimensions. We overcome this limitation by considering the aggregation of identically distributed risks with generalized FGM copula specified by a common parameter $p$. In this case, the analogy with the geometrical structure of the class of Bernoulli distribution allows us to provide sharp analytical bounds for convex risk measures.

q-fin.MF↗

arXiv2024

Collective risk models with FGM dependence

We study copula-based collective risk models when the dependence structure is defined by a Farlie-Gumbel-Morgenstern (FGM) copula. By leveraging a one-to-one correspondence between the class of FGM copulas and multivariate symmetric Bernoulli distributions, we find convenient representations for the moments and Laplace-Stieltjes transform for the aggregate random variable defined from collective risk models with FGM dependence. We examine different components of this collective risk model, aiming to better understand the impact of the assumed dependence between a claim's frequency and severity. Relying on stochastic ordering, we analyze the impact of dependence on the aggregate claim amount random variable. Even if the FGM copula may only induce moderate dependence, we illustrate through numerical examples that the cumulative effect of FGM dependence can lead to substantial variations in key risk measures on aggregate random variables defined from collective risk models.

stat.AP↗

arXiv2024

Parametric estimation of conditional Archimedean copula generators for censored data

In this paper, we propose a novel approach for estimating Archimedean copula generators in a conditional setting, incorporating endogenous variables. Our method allows for the evaluation of the impact of the different levels of covariates on both the strength and shape of dependence by directly estimating the generator function rather than the copula itself. As such, we contribute to relaxing the simplifying assumption inherent in traditional copula modeling. We demonstrate the effectiveness of our methodology through applications in two diverse settings: a diabetic retinopathy study and a claims reserving analysis. In both cases, we show how considering the influence of covariates enables a more accurate capture of the underlying dependence structure in the data, thus enhancing the applicability of copula models, particularly in actuarial contexts.

stat.ME↗

arXiv2024

A non-parametric estimator for Archimedean copulas under flexible censoring scenarios and an application to claims reserving

With insurers benefiting from ever-larger amounts of data of increasing complexity, we explore a data-driven method to model dependence within multilevel claims in this paper. More specifically, we start from a non-parametric estimator for Archimedean copula generators introduced by Genest and Rivest (1993), and we extend it to diverse flexible censoring scenarios using techniques derived from survival analysis. We implement a graphical selection procedure for copulas that we validate using goodness-of-fit methods applied to complete, single-censored, and double-censored bivariate data. We illustrate the performance of our model with multiple simulation studies. We then apply our methodology to a recent Canadian automobile insurance dataset where we seek to model the dependence between the activation delays of correlated coverages. We show that our model performs quite well in selecting the best-fitted copula for the data at hand, especially when the dataset is large, and that the results can then be used as part of a larger claims reserving methodology.

stat.ME↗

arXiv2023

Individual Claims Reserving using Activation Patterns

The occurrence of a claim often impacts not one but multiple insurance coverages provided in the contract. To account for this multivariate feature, we propose a new individual claims reserving model built around the activation of the different coverages to predict the reserve amounts. Using the framework of multinomial logistic regression, we model the activation of the different insurance coverages for each claim and their development in the following years, i.e. the activation of other coverages in the later years and all the possible payments that might result from them. As such, the model allows us to complete the individual development of the open claims in the portfolio. Using a recent automobile dataset from a major Canadian insurance company, we demonstrate that this approach generates accurate predictions of the total reserves as well as of the reserves per insurance coverage. This allows the insurer to get better insights in the dynamics of his claims reserves.

q-fin.MF↗

arXiv2022

Preventing RNN from Using Sequence Length as a Feature

Recurrent neural networks are deep learning topologies that can be trained to classify long documents. However, in our recent work, we found a critical problem with these cells: they can use the length differences between texts of different classes as a prominent classification feature. This has the effect of producing models that are brittle and fragile to concept drift, can provide misleading performances and are trivially explainable regardless of text content. This paper illustrates the problem using synthetic and real-world data and provides a simple solution using weight decay regularization.

cs.LG↗

arXiv2022

A new method to construct high-dimensional copulas with Bernoulli and Coxian-2 distributions

We propose an approach to construct a new family of generalized Farlie-Gumbel-Morgenstern (GFGM) copulas that naturally scales to high dimensions. A GFGM copula can model moderate positive and negative dependence, cover different types of asymmetries, and admits exact expressions for many quantities of interest such as measures of association or risk measures in actuarial science or quantitative risk management. More importantly, this paper presents a new method to construct high-dimensional copulas based on mixtures of power functions, and may be adapted to more general contexts to construct broader families of copulas. We construct a family of copulas through a stochastic representation based on multivariate Bernoulli distributions and Coxian-2 distributions. This paper will cover the construction of a GFGM copula, study its measures of multivariate association and dependence properties. We explain how to sample random vectors from the new family of copulas in high dimensions. Then, we study the bivariate case in detail and find that our construction leads to an asymmetric modified Huang-Kotz FGM copula. Finally, we study the exchangeable case and provide some insights into the most negative dependence structure within this new class of high-dimensional copulas.

math.ST↗

arXiv2022

Risk aggregation with FGM copulas

We offer a new perspective on risk aggregation with FGM copulas. Along the way, we discover new results and revisit existing ones, providing simpler formulas than one can find in the existing literature. This paper builds on two novel representations of FGM copulas based on symmetric multivariate Bernoulli distributions and order statistics. First, we detail families of multivariate distributions with closed-form solutions for the cumulative distribution function or moments of the aggregate random variables. We order aggregate random variables under the convex order and provide methods to compute the cumulative distribution function of aggregate rvs when the marginals are discrete. Finally, we discuss risk-sharing and capital allocation, providing numerical examples for each.

math.ST↗

arXiv2022

Exchangeable FGM copulas

Copulas are a powerful tool to model dependence between the components of a random vector. One well-known class of copulas when working in two dimensions is the Farlie-GumbelMorgenstern (FGM) copula since their simple analytic shape enables closed-form solutions to many problems in applied probability. However, the classical definition of high-dimensional FGM copula does not enable a straightforward understanding of the effect of the copula parameters on the dependence, nor a geometric understanding of their admissible range. We circumvent this issue by studying the FGM copula from a probabilistic approach based on multivariate Bernoulli distributions. This paper studies high-dimensional exchangeable FGM copulas, a subclass of FGM copulas. We show that dependence parameters of exchangeable FGM can be expressed as convex hulls of a finite number of extreme points and establish partial orders for different exchangeable FGM copulas (including maximal and minimal dependence). We also leverage the probabilistic interpretation to develop efficient sampling and estimating procedures and provide a simulation study. Throughout, we discover geometric interpretations of the copula parameters that assist one in decoding the dependence of high-dimensional exchangeable FGM copulas.

math.ST↗

arXiv2022

Micro-level Reserving for General Insurance Claims using a Long Short-Term Memory Network

Detailed information about individual claims are completely ignored when insurance claims data are aggregated and structured in development triangles for loss reserving. In the hope of extracting predictive power from the individual claims characteristics, researchers have recently proposed to move away from these macro-level methods in favor of micro-level loss reserving approaches. We introduce a discrete-time individual reserving framework incorporating granular information in a deep learning approach named Long Short-Term Memory (LSTM) neural network. At each time period, the network has two tasks: first, classifying whether there is a payment or a recovery, and second, predicting the corresponding non-zero amount, if any. We illustrate the estimation procedure on a simulated and a real general insurance dataset. We compare our approach with the chain-ladder aggregate method using the predictive outstanding loss estimates and their actual values. Based on a generalized Pareto model for excess payments over a threshold, we adjust the LSTM reserve prediction to account for extreme payments.

cs.LG↗

arXiv2021

Geographic ratemaking with spatial embeddings

Spatial data is a rich source of information for actuarial applications: knowledge of a risk's location could improve an insurance company's ratemaking, reserving or risk management processes. Insurance companies with high exposures in a territory typically have a competitive advantage since they may use historical losses in a region to model spatial risk non-parametrically. Relying on geographic losses is problematic for areas where past loss data is unavailable. This paper presents a method based on data (instead of smoothing historical insurance claim losses) to construct a geographic ratemaking model. In particular, we construct spatial features within a complex representation model, then use the features as inputs to a simpler predictive model (like a generalized linear model). Our approach generates predictions with smaller bias and smaller variance than other spatial interpolation models such as bivariate splines in most situations. This method also enables us to generate rates in territories with no historical experience.

stat.AP↗