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We investigate in this work the K-sample clustering of populations issued from contamination phenomenon. A contamination model is a two-component mixture model in which one component is known (standard behaviour) when the second one, modelling a departure from the standard behaviour, is unknown. When K populations from such a model are observed we propose a semiparametric clustering methodology to detect, for coordinated diagnosis and/or best practices sharing purpose, which populations are impacted by the same type of contamination. We prove the consistency of our approach under the existence of true clusters and show the performances of our methodology through an extensive Monte Carlo study. We finally apply our methodology, implemented in the admix R package, to a European countries COVID-19 excess of mortality dataset for which we aim to cluster countries similarly impacted by the pandemic over classes of age.

International audience; In this paper, we consider two-component mixture models having one single 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 test the equality in distribution of the unknown random sources involved in two separate samples generated from such a model. For this purpose, we introduce the so-called IBM (Inversion-Best Matching) approach resulting in a tuning-free relaxed semiparametric Cramér-von Mises type two-sample test requiring minimal assumptions about the unknown distributions. The accomplishment of our work lies in the fact that we establish, under some natural and interpretable mutual-identifiability conditions specific to the two-sample case, a functional central limit theorem about the proportion parameters along with the unknown cumulative distribution functions of the model. 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, implemented in the admix R package, is applied to a real-life situation through pairwise post COVID-19 mortality excess profile testing across a panel of European countries.

We propose a clustering procedure to group K populations into subgroups with the same dependence structure. The method is adapted to paired population and can be used with panel data. It relies on the differences between orthogonal projection coefficients of the K density copulas estimated from the K populations. Each cluster is then constituted by populations having significantly similar dependence structures.A recent test statistic from Ngounou-Bakam and Pommeret (2022) is used to construct automatically such clusters. The procedure is data driven and depends on the asymptotic level of the test. We illustrate our clustering algorithm via numerical studies and through two real datasets: a panel of financial datasets and insurance dataset of losses and allocated loss adjustment expense.

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., 42 pages, 6 figures, 11 tables

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., Nous considérons dans cet article des mélanges à deux composantes dont une composante est connue. Ces modèles sont d'un intérêt particulier lorsqu'un phénomène bien connu est affecté par une perturbation inconnue. Nous proposons de comparer les sources de cet impact inconnu entre deux échantillons distincts. Pour ce faire nous introduisons une nouvelle approche, appelée IBM (Inversion-Best Matching), qui permet d'effectuer le test à la manière d'un test de type Cramer-Von Mises sous très peu d'hypothèses concernant les aléas perturbateurs inconnus. Nous établissons un théorème central limite (TCL) fonctionnel sur les estimateurs que sont les proportions de chacun des modèles de contamination, ainsi que la fonction de répartition inconnue, là où Patra et Sen [22] n'obtenait pas de résultat asymptotique de type TCL. Des simulations permettent de valider la performance de la procédure de test et d'en valider ses propriétés asymptotiques. Enfin, nous appliquons notre stratégie de test à un problème concret de comparaison de déformation de la mortalité suite à la pandémie de Covid-19 à travers plusieurs pays européens.

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.

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. (2006) and comment our results. Finally we discuss possible extension of this work to more general models. MSC 2010 subject classifications: Primary 62F03, 28C20; secondary 33C45.

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We consider the problem of missing data when the mechanism of missingness is not at random and when the partially observed vari- able has known or observed moments. A nonparametric estimator of the probability of missingness is proposed. A data driven statistic is constructed to test the missingness mechanism. Illustrations through univariate logistic regressions are presented: the method permits to estimate regression coe cients when the covariate is completely miss- ing for one response category. A test of signi cance is proposed for the coe cients. The performance of the method is investigated in a simulation study. An illustration is considered using a real data set.

In the Bayesian stochastic search variable selection framework, a common prior distribution for the regression coefficients is the g-prior of Zellner (1986). However, there are two standard cases in which the associated covariance matrix does not exist, and the conventional prior of Zellner can not be used: if the number of observations is lower than the number of variables (large p and small n paradigm), or if some variables are linear combinations of others. In such situations a prior distribution derived from the prior of Zellner can be used, by introducing a ridge parameter. This prior introduced by Gupta and Ibrahim (2007) is a flexible and simple adaptation of the g-prior. In this paper we study the influence of the ridge parameter on the selection of variables. A simple way to choose the associated hyper-parameters is proposed. The method is valid for any generalized linear mixed model and we focus on the case of probit mixed models when some variables are linear combinations of others. The method is applied to both simulated and real datasets obtained from Affymetrix microarray experiments. Results are compared to those obtained with the Bayesian Lasso.

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Consider two random variables contaminated by two unknown transformations. The aim of this paper is to test the equality of those transformations. Two cases are distinguished: first, the two random variables have known distributions. Second, they are unknown but observed before contaminations. We propose a nonparametric test statistic based on empirical cumulative distribution functions. Monte Carlo studies are performed to analyze the level and the power of the test. An illustration is presented through a real data set., 15 pages

The problem of handling missing data for a linear mixed model in presence of correlation between covariates is considered. The missing mechanism concerns both dependent variable and design matrix. We propose an imputation algorithm combining multiple imputation and Partial Least Squares (PLS) analysis methods. Our method relies on two steps: removing random effects, fixed effects are first imputed and PLS components are constructed on the corresponding complete case. The dependent variable is then imputed inside the linear mixed model built by adding the random effects to PLS components. The method is applied on simulations and on real data.

We consider the problem of testing whether two samples of contaminated data, possibly paired, are from the same distribution. Is is assumed that the contaminations are additive noises with known moments of all orders. The test statistic is based on the polynomials moments of the difference between observations and noises. . 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. A real-data example is presented to demonstrate the method.

The problem of handling missing data in generalized linear mixed models with correlated covariates is considered when the missing mechanism concerns both the response variable and the covariates. An imputation algorithm combining multiple imputation and Partial Least Squares (PLS) regression is proposed. The method relies on two steps. In a first step, using a linearization technique, the generalized linear mixed model is approximated by a linear mixed model. A latent variable is introduced and its associated PLS components are constructed. In a second step these PLS components are used in the generalized linear mixed model to impute the response variable. The method is applied on simulations and on a real data.

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International audience; En présence d'une variable de loi connue dont les données manquantes ne sont pas dues au hasard (No Missing At Random) nous proposons une méthode pour estimer les probabilités d'avoir une donnée manquante suivant les valeurs observées de la variable. Nous en déduisons une statistique pour tester si les données manquent au hasard ou non.

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 propose a smooth test of comparison of two distribution functions. This test adapts to the classical two-sample problem as well as that of paired populations, including discrete distributions. A simulation study and an application to real data show its good performances.