arXiv subjects

Christian Y. Robert

Publications and source records attributed to Christian Y. Robert.

16 recordsLinked to original sources

arXiv2026

Linear Risk Sharing in Community-Based Insurance: Ruin Reduction in the Compound Poisson Model

This paper studies proportional risk sharing at claim occurrence time in community-based insurance. Each participant is modeled by an individual Cramér-Lundberg surplus process, and, whenever a claim is reported within the pool, its cost is redistributed according to a fixed allocation matrix. We compare the infinite-time ruin probability of each participant under stand-alone operation and under pool participation. Our main result shows that pooling reduces, for every participant, the infinite-time ruin probability when claim severities belong to a common scale family, the allocation rule satisfies full allocation and actuarial fairness, and each transfer remains bounded by an individual capacity condition. The proof relies on a convex-order comparison between the losses borne inside the pool and the corresponding stand-alone losses. We also clarify the role of these assumptions by showing that, outside this framework, pooling need not be beneficial for all participants. Numerical illustrations with Exponential and LogNormal severities support the theoretical findings and highlight how the design of proportional sharing rules affects solvency. The paper thus provides simple and interpretable sufficient conditions under which transparent linear risk-sharing arrangements improve individual solvency in community-based insurance.

math.PR↗

arXiv2026

Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$

An isotropic fractional Brownian field (with Hurst parameter $H<1/2$) is observed in a family of points in the unit square $\mathbf{C}=(-1/2,1/2]^{2}$% . These points are assumed to come from a realization of a homogeneous Poisson point process with intensity $N$. We consider normalized increments (resp. pairs of increments) along the edges of the Delaunay triangulation generated by the Poisson point process (resp. pairs of edges within triangles). Central limit theorems are established for the respective centered squared increment sums as $N\rightarrow \infty $.

math.PR↗

arXiv2026

Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics

We study squared increment sums of the pointwise maximum of two independent and identically distributed isotropic fractional Brownian fields over a fixed two-dimensional domain. The fields are observed at the points of a homogeneous Poisson point process with intensity $N$, and increments are computed along the edges of the associated Delaunay triangulation. In contrast with the case of a single fractional Brownian field, where centered squared increment sums satisfy a central limit theorem after the usual normalization, the pointwise maximum exhibits a different asymptotic regime. The dominant contribution comes from Delaunay edges located in a shrinking neighborhood of the random interface where the two fractional Brownian fields exchange the role of the maximizer. For Hurst parameter $H<1/2$, we prove that the properly normalized squared increment sum converges in probability to a deterministic constant times the local time at zero of the difference between the two fields. This shows that the asymptotic behavior is governed by the geometry of the random contact set rather than by Gaussian fluctuation effects. The result provides a key ingredient for fixed-domain asymptotic inference in Brown--Resnick type models based on randomly located observations.

math.PR↗

arXiv2026

Asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields over a fixed-domain

Likelihood-based inference for max-stable random fields is challenging, since finite-dimensional densities are either unavailable in closed form or computationally intractable in moderate to high dimension. Composite likelihood methods, based on low-dimensional marginal densities, therefore provide a natural alternative. In this paper, we study maximum composite likelihood estimation for spatial Brown--Resnick random fields generated by isotropic fractional Brownian fields. We work under fixed-domain asymptotics: a single realization of the max-stable field is observed on an increasingly dense random set of sites, given by a homogeneous Poisson point process. Pairwise and triplewise composite likelihoods are constructed by retaining, respectively, the edges and the triangles of the associated Poisson--Delaunay triangulation. Our main results establish the consistency of the resulting maximum composite likelihood estimators of the scale and smoothness parameters, when the other parameter is known. Their asymptotic behaviour is non-standard: the estimators converge at rates depending on the smoothness parameter and their centered limits are non-Gaussian. More precisely, the limiting fluctuations are driven by aggregated local times associated with the canonical tessellation of the Brown--Resnick field. These results reveal a fundamental departure from the classical composite likelihood theory based on increasing domains or independent replications, and show that Gaussian uncertainty quantification may be misleading in fixed-domain inference for max-stable spatial extremes.

math.ST↗

arXiv2022

Fixed-domain asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields

Likelihood inference for max-stable random fields is in general impossible because their finite-dimen\-sional probability density functions are unknown or cannot be computed efficiently. The weighted composite likelihood approach that utilizes lower dimensional marginal likelihoods (typically pairs or triples of sites that are not too distant) is rather favored. In this paper, we consider the family of spatial max-stable Brown-Resnick random fields associated with isotropic fractional Brownian fields. We assume that the sites are given by only one realization of a homogeneous Poisson point process restricted to $\mathbf{C}=(-1/2,1/2]^{2}$ and that the random field is observed at these sites. As the intensity increases, we study the asymptotic properties of the composite likelihood estimators of the scale and Hurst parameters of the fractional Brownian fields using different weighting strategies: we exclude either pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles.

math.ST↗

arXiv2020

Stochastic derivative estimation for max-stable random fields

We consider expected performances based on max-stable random fields and we are interested in their derivatives with respect to the spatial dependence parameters of those fields. Max-stable fields, such as the Brown--Resnick and Smith fields, are very popular in spatial extremes. We focus on the two most popular unbiased stochastic derivative estimation approaches: the likelihood ratio method (LRM) and the infinitesimal perturbation analysis (IPA). LRM requires the multivariate density of the max-stable field to be explicit, and IPA necessitates the computation of the derivative with respect to the parameters for each simulated value. We propose convenient and tractable conditions ensuring the validity of LRM and IPA in the cases of the Brown--Resnick and Smith field, respectively. Obtaining such conditions is intricate owing to the very structure of max-stable fields. Then we focus on risk and dependence measures, which constitute one of the several frameworks where our theoretical results can be useful. We perform a simulation study which shows that both LRM and IPA perform well in various configurations, and provide a real case study that is valuable for the insurance industry.

q-fin.RM↗

arXiv2019

Power variations for a class of Brown-Resnick processes

We consider the class of simple Brown-Resnick max-stable processes whose spectral processes are continuous exponential martingales. We develop the asymptotic theory for the realized power variations of these max-stable processes, that is, sums of powers of absolute increments. We consider an infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed. More specifically we obtain a biased central limit theorem whose bias depend on the local times of the differences between the logarithms of the underlying spectral processes.

math.ST↗

arXiv2018

A central limit theorem for functions of stationary max-stable random fields on $\mathbb{R}^d$

Max-stable random fields are very appropriate for the statistical modelling of spatial extremes. Hence, integrals of functions of max-stable random fields over a given region can play a key role in the assessment of the risk of natural disasters, meaning that it is relevant to improve our understanding of their probabilistic behaviour. For this purpose, in this paper, we propose a general central limit theorem for functions of stationary max-stable random fields on $\mathbb{R}^d$. Then, we show that appropriate functions of the Brown-Resnick random field with a power variogram and of the Smith random field satisfy the central limit theorem. Another strong motivation for our work lies in the fact that central limit theorems for random fields on $\mathbb{R}^d$ have been barely considered in the literature. As an application, we briefly show the usefulness of our results in a risk assessment context.

math.PR↗

arXiv2018

Geometric ergodicity for some space-time max-stable Markov chains

Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010).

math.PR↗

arXiv2014

Likelihood based inference for high-dimensional extreme value distributions

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail-dependence. In areas such as the modeling of financial and insurance risks, or as the modeling of spatial variables, extreme value models in high dimensions (up to fifty or more) with their statistical inference procedures are needed. In this paper, we consider max-stable models for which the spectral random vectors have absolutely continuous distributions. For random samples with max-stable distributions we provide quasi-explicit analytical expressions of the full likelihoods. When the full likelihood becomes numerically intractable because of a too large dimension, it is however necessary to split the components into subgroups and to consider a composite likelihood approach. For random samples in the max-domain of attraction of a max-stable distribution, two approaches that use simpler likelihoods are possible: (i) a threshold approach that is combined with a censoring scheme, (ii) a block maxima approach that exploits the information on the occurrence times of the componentwise maxima. The asymptotic properties of the estimators are given and the utility of the methods is examined via simulation. The estimators are also compared with those derived from the pairwise composite likelihood method which has been previously proposed in the spatial extreme value literature.

stat.AP↗

arXiv2013

Estimating the efficient price from the order flow: a Brownian Cox process approach

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price,...). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price form the order flow.

q-fin.TR↗

arXiv2010

Subsampling weakly dependent times series and application to extremes

This paper provides extensions of the work on subsampling by Bertail et al. (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi (1999). We investigate properties of smooth and rough subsampling estimators for distributions of converging and extreme statistics when the underlying time series is η or λ-weakly dependent.

math.ST↗

arXiv2009

Inference for the limiting cluster size distribution of extreme values

Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data.

stat.AP↗

arXiv2008

A Sliding Blocks Estimator for the Extremal Index

In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on the estimation of the extremal index which measures the degree of clustering of extremes. We consider disjoint and sliding blocks estimators and compare their asymptotic properties. In particular we show that the sliding blocks estimator is more efficient than the disjoint version and has a smaller asymptotic bias. Moreover we propose a method to reduce its bias when considering sufficiently large block sizes.

math.ST↗

arXiv2008

Estimating the multivariate extremal index function

The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study.

stat.AP↗

arXiv2007

Tails of random sums of a heavy-tailed number of light-tailed terms

The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a random number of claims. If the tail of the claim number is heavier than the tail of the claim sizes, then under certain conditions the tail of the total claim size does not change asymptotically if the individual claim sizes are replaced by their expectations. The conditions allow the claim number distribution to be of consistent variation or to be in the domain of attraction of a Gumbel distribution with a mean excess function that grows to infinity sufficiently fast. Moreover, the claim number is not necessarily required to be independent of the claim sizes.

math.PR↗