POMMERET Denys

In this paper, we consider two-component mixture distributions having one known component. This type of model is of particular interest when a known random phenomenon is contaminated by an unknown random effect. We propose in this setup to compare the unknown random sources involved in two separate samples. For this purpose, we introduce the so-called IBM (Inversion-Best Matching) approach resulting in a relaxed semiparametric Cramér-von Mises type two-sample test requiring very minimal assumptions (shape constraint free) about the unknown distributions. The accomplishment of our work lies in the fact that we establish a functional central limit theorem on the proportion parameters along with the unknown cumulative distribution functions of the model when Patra and Sen [22] prove that the √ n-rate cannot be achieved on these quantities in the basic one-sample case. An intensive numerical study is carried out from a large range of simulation setups to illustrate the asymptotic properties of our test. Finally, our testing procedure is applied to a real-life application through pairwise post-covid mortality effect testing across a panel of European countries.

In this paper we study nonparametric estimators of copulas and copula densities. We first focus our study on a density copula estimator based on a polynomial orthogonal projection of the joint density. A new copula estimator is then deduced. Its asymptotic properties are studied: we provide a large functional class for which this construction is optimal in the minimax and maxiset sense and we propose a method selection for the smoothing parameter. An intensive simulation study shows the very good performance of both copulas and copula densities estimators which we compare to a large panel of competitors. A real dataset in actuarial science illustrates this approach.

We consider in this paper two-component mixture distributions having one known component. This is the case when a gold standard reference component is well known, and when a population contains such a component plus another one with different features. When two populations are drawn from such models, we propose a penalized Chi-squared type testing procedure able to compare pairwise the unknown components, i.e. to test the equality of their residual features densities. An intensive numerical study is carried out from a large range of simulation setups to illustrate the asymptotic properties of our test. Moreover the testing procedure is applied on two real cases: i) mortality datasets, where results show that the test remains robust even in challenging situations where the unknown component only represents a small percentage of the global population, ii) galaxy velocities datasets, where stars luminosity mixed with the Milky Way are compared.

Discrepancies in population structures, decision making, health systems and numerous other factors result in various COVID-19-mortality dynamics at country scale, and make the forecast of deaths in a country under focus challenging. However, mortality dynamics of countries that are ahead of time implicitly include these factors and can be used as real-life competing predicting models. We precisely propose such a data-driven approach implemented in a publicly available web app timely providing mortality curves comparisons and real-time short-term forecasts for about 100 countries. Here, the approach is applied to compare the mortality trajectories of second-line and front-line European countries facing the COVID-19 epidemic wave. Using data up to mid-April, we show that the second-line countries generally followed relatively mild mortality curves rather than fast and severe ones. Thus, the continuation, after mid-April, of the COVID-19 wave across Europe was likely to be mitigated and not as strong as it was in most of the front-line countries first impacted by the wave (this prediction is corroborated by posterior data).

Measurements are an everyday act, they give us a lot of information and allow us to make decisions. The analysis of measurements can allow us to learn more about our environment, but the error of a measurement can have important consequences in certain fields. In a first part, we propose, thanks to the study of blood analysis measurements carried out at the University Hospital of Clermont-Ferrand, a procedure allowing to detect the drifts of the analyzers of medical biology laboratories, based on the measurements of patients' analyses. After a descriptive analysis of the data, the method implemented, using methods for detecting breaks in time series, is tested for simulations of breaks representing shifts, inaccuracies or drifts of analyzers for different biological parameters measured. The method is adapted for two scenarios: when the patients' hospital service is known or not. The study is completed by an analysis of the impact of the measurement uncertainty on the patients' analyses. In a second part we study measurements of volcanic ash shapes made at the Magmas and Volcanoes Laboratory of the University of Clermont Auvergne, in order to determine a link between the collection locations and the particle shapes. After having shown the dependence between these parameters, we propose, thanks to a classification method, a grouping of the particles representing different populations depending on the distance between the collection sites and the crater of the volcano.

In October 2014, the French Environment and Energy Management Agency (ADEME) in cooperation with the company ENEDIS (formerly ERDF for Électricité Réseau Distribution France) started a research project called "smart-grid SOLidarity-ENergy-iNovation" (SOLENN) with the objectives of studying the control of electricity consumption by supporting households and securing the electricity supply among others. This thesis is part of the above-mentioned objectives. The SOLENN project is piloted by the ADEME and took place in the town of Lorient. The aim of the project is to implement a pedagogy to make households aware of energy saving. In this context, we discuss a method for estimating extreme quantiles and rare event probabilities for non-parametric functional data which is the subject of an R package. We then propose an extension of the famous Cox proportional hazards model and allow the estimation of rare event probabilities and extreme quantiles. Finally, we give the application of some statistical models developed in this paper on electricity consumption data and which proved to be useful for the SOLENN project. A first application is in connection with the shaving program led by ENEDIS in order to secure the operation of the electrical network. A second application is the implementation of the linear model to study the effect of several individual visits on the electricity consumption.

In this paper we investigate a semiparametric testing approach to answer if the Gaussian assumption made by McLachlan et al. (2006) on the unknown component of their false discovery type mixture model was a posteriori correct or not. Based on a semiparametric estimation of the Eu-clidean parameters of the model (free from the Gaussian assumption), our method compares pairwise the Hermite coefficients of the model estimated directly from the data with the ones obtained by plugging the estimated parameters into the Gaussian version of the false discovery mixture model. These comparisons are incorporated into a sum of square type statistic which order is controlled by a penalization rule. We prove under mild conditions that our test statistic is asymptotically χ 2 (1)-distributed and study its behavior under different types of alternatives, including contiguous non-parametric alternatives. Several level and power studies are numerically conducted on models close to those considered in McLachlan et al. (2006) to validate the suitability of our approach. We also discuss the lack of power of the maximum likelihood version of our test in a neighborhood of certain non identifiable situations and implement our testing procedure on the three microarray real datasets analyzed in McLachlan et al.

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This article proposes a parsimonious alternative approach for modeling the stochastic dynamics of mortality rates. Instead of the commonly used factor-based decomposition framework , we consider modeling mortality improvements using a random field specification with a given causal structure. Such a class of models introduces dependencies among adjacent cohorts aiming at capturing, among others, the cohort effects and cross generations correlations. It also describes the conditional heteroskedasticity of mortality. The proposed model is a generalization of the now widely used AR-ARCH models for random processes. For such class of models, we propose an estimation procedure for the parameters. Formally, we use the quasi-maximum likelihood estimator (QMLE) and show its statistical consistency and the asymptotic normality of the estimated parameters. The framework being general, we investigate and illustrate a simple variant, called the three-level memory model, in order to fully understand and assess the effectiveness of the approach for modeling mortality dynamics.

All marine ecosystems rely on phytoplankton to convert atmospheric carbon into organic matter through the process of photosynthesis. Two approaches are presented to measure phytoplankton productivity by taking into account the temporal evolution of cell size. In essence, they reflect the progressive assimilation of inorganic carbon during the life cycle of a cell and its reallocation from one generation to the next at the time of cell division. In the natural environment, this carbon flow depends on phytoplankton communities and their sensitivities. The observation of phytoplankton in a disturbed environment on a small scale of time and/or space is essential to anticipate climate change. The Mediterranean Sea in particular is likely to undergo rapid changes in its climate and the populations it supports. In the Mediterranean Sea as in the global ocean, measurement campaigns are the basis of scenarios that reflect the impact of the environment on the functioning and capacity of phytoplankton to buffer gas emissions from human activity.

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A numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model. The defective density function associated to the ruin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measure that belongs to Natural Exponential Family with Quadratic Variance Function (NEF-QVF). The method is convenient in at least four ways. Firstly, it leads to a simple analytical expression of the ultimate ruin probability. Secondly, the implementation does not require strong computer skills. Thirdly, our approximation method does not necessitate any preliminary discretisation step of the claim sizes distribution. Finally, the coefficients of our formula do not depend on initial reserves.

This thesis work focuses on various characterizations of multivariate models of multiple stable-Tweedie in the context of natural exponential families under the steepness property. These models appeared in 2014 in the literature were first introduced and described in a restricted form of normal stable-Tweedie before extensions to multiple cases. They consist of a mixture of a one-dimensional stable-Tweedie law of fixed positive real variable, and stable-Tweedie laws of independent real variables conditional on the first fixed one, with the same variance equal to the value of the fixed variable. The corresponding normal stable-Tweedie models are those of the mixture of a fixed positive one-dimensional stable-Tweedie law and the others all independent Gaussian. Through special cases such as normal, Poisson, gamma, inverse Gaussian, multiple stable-Tweedie models are very common in applied statistics and probability studies. First, we have characterized the normal stable-Tweedie models through their variance functions or covariance matrices expressed in terms of their mean vectors. The nature of the polynomials associated with these models is deduced according to the values of the variance power using the properties of quasi-orthogonality, Lévy-Sheffer systems, and polynomial recurrence relations. Then, these first results allowed us to characterize the largest class of multiple stable-Tweedie using the variance function. This led to a new classification which makes the family much more understandable. Finally, an extension of the characterization of normal stable-Tweedie by generalized variance function or determinant of the variance function has been established via their indefinite divisibility property and by passing through the corresponding Monge-Ampere equations. Expressed as the product of the components of the mean vector in multiple powers, the characterization of all multivariate stable-Multiple Weedie models by generalized variance function remains an open problem.

It is frequent that observations arise from a random variable modified by an unknown transformation. This problem is considered in a two-sample context when two random variables are perturbed by two unknown transformations. We propose a test for the equality of those transformations. Two cases are considered: first, the two random variables have known distributions. Second, they have unknown distributions but they are observed before transformations. We propose nonparametric test statistics based on empirical cumulative distribution functions. In the first case the asymptotic distribution of the test statistic is the standard normal distribution. In the second case it is shown that the asymptotic distribution is a convolution of exponential distributions. The convergence under contiguous alternatives is studied. Monte Carlo studies are performed to analyze the level and the power of the test. An illustration is presented through a real data set.

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A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.

Abstract In this paper we consider two nonparametric mixtures of quadratic natural exponential families with unknown mixing densities. We propose a statistic to test the equality of these mixing densities when the two natural exponential families are known. The test is based on moment characterizations of the distributions. The number of moments is retained automatically by a data driven technique. Some examples and simulations of implementation of the procedure are provided.

The purpose of this thesis is to study numerical methods for approximating the probability density associated with random variables that have compound distributions. These random variables are commonly used in actuarial science to model the risk borne by a portfolio of contracts. In catastrophe theory, the probability of ultimate catastrophe in the compound Poisson model is equal to the survival function of a compound geometric distribution. The proposed numerical method consists of an orthogonal projection of the density onto a basis of orthogonal polynomials. These polynomials are orthogonal with respect to a reference probability measure belonging to the Quadratic Natural Exponential Families. The polynomial approximation method is compared to other density approximation methods based on moments and the Laplace transform of the distribution. The extension of the method in dimension higher than $1$ is presented, as well as the obtention of a density estimator from the approximation formula. This thesis also includes the description of an aggregation method adapted to portfolios of life insurance contracts of individual savings type. The aggregation procedure leads to the construction of model points to allow the evaluation of the best estimate reserves in a reasonable time and in accordance with the European Solvency II directive.

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The Equi-Energy Sampler (EES) introduced by Kou et al [2006] is based on a population of chains which are updated by local moves and global moves, also called equi-energy jumps. The state space is partitioned into energy rings, and the current state of a chain can jump to a past state of an adjacent chain that has energy level close to its level. This algorithm has been developed to facilitate global moves between different chains, resulting in a good exploration of the state space by the target chain. This method seems to be more efficient than the classical Parallel Tempering (PT) algorithm. However it is difficult to use in combination with a Gibbs sampler and it necessitates increased storage. In this paper we propose an adaptation of this EES that combines PT with the principle of swapping between chains with same levels of energy. This adaptation, that we shall call Parallel Tempering with Equi-Energy Moves (PTEEM), keeps the original idea of the EES method while ensuring good theoretical properties, and practical implementation even if combined with a Gibbs sampler. Performances of the PTEEM algorithm are compared with those of the EES and of the standard PT algorithms in the context of mixture models, and in a problem of identification of gene regulatory binding motifs.

Approximate Bayesian Computational (ABC) methods (or likelihood-free methods) have appeared in the past fifteen years as useful methods to perform Bayesian analyses when the likelihood is analytically or computationally intractable. Several ABC methods have been proposed: Monte Carlo Markov Chains (MCMC) methods have been developped by Marjoramet al. (2003) and by Bortotet al. (2007) for instance, and sequential methods have been proposed among others by Sissonet al.

This paper considers two random variables such that there exists a monotone transformation between their distribution functions. The problem is to test if there is a change in this transformation when these two variables are observed under K different conditions. The approach considered is a CUSUM test based on the cumulative sum of the residuals and a test statistic is proposed for testing the equality of the K transformations. The asymptotic distribution of the test statistic is derived and its finite sample properties are examined by simulation. As a further illustration, an analysis of a real data set concerning the impact of the financial crisis of September 2008 is given.

In this paper we consider the problem of testing whether two samples of contaminated data arise from the same distribution. Is is assumed that the contaminations are additive noises with known, or estimated moments. This situation can also be viewed as two signals observed before and after perturbations. The problem is then to test the equality of both perturbations. The test statistic is based on the polynomials moments of the difference between observations and noises. The test is very simple and allows one to compare two independent as well as two paired contaminated samples. A data driven selection is proposed to choose automatically the number of involved polynomials. We present a simulation study in order to investigate the power of the proposed test within discrete and continuous cases. Real-data examples are presented to illustrate the method. Copyright Springer-Verlag Berlin Heidelberg 2013.

The issue of surrender has long been of concern to insurers, particularly in the context of life insurance savings contracts, for which colossal sums are at stake. The emergence of the European Solvency II directive, which recommends the development of internal models (of which an entire module is dedicated to the management of surrender behavior risks), reinforces the need to deepen our knowledge and understanding of this risk. It is in this context that we address in this thesis the issues of segmentation and modeling of surrenders, with the objective of better understanding and taking into account all the key factors that influence policyholders' decisions. The heterogeneity of behaviors and their correlation, as well as the environment to which policyholders are subjected, are as many difficulties to be treated in a specific way in order to make forecasts. We have developed a methodology that has produced very encouraging results and has the advantage of being replicable by adapting it to the specificities of different product lines. Through this modeling, model selection appears as a central point. We address it by establishing the strong convergence properties of a new estimator, as well as the consistency of a new selection criterion in the context of mixtures of generalized linear models.

This work proposes the generalization to several dimensions of different characterizations of the class of real quadratic natural exponential families, using the theory of orthogonal polynomials. In a first part, written in English and submitted for publication, we develop the three following characterizations: i) of meixner (1934) concerning the orthogonal polynomials of exponential generating function, ii) of feinsilver (1986) where the polynomials are obtained by derivations of the densities of probabilities, iii) of shanbhag (1972) where appears the bhattacharrya matrices. By introducing an original construction of orthogonal polynomials, we obtain a characterization of natural quadratic and simple quadratic multidimensional exponential families. Moreover, we determine the class of orthogonal polynomials in several variables whose generating function is exponential. The second part is inspired by an article of feinsilver (1991) which shows a link between the algebra of links and the theory of probabilities. Based on this work, we then show the existence of a bijection between the class of simple quadratic natural exponential families and three types of Lie algebras. Thus, any probability of a natural exponential family allows to define operators of one of the three lie algebras in question thanks to the recurrence equations of the orthogonal polynomials considered in the first part.