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# Bilodeau, Martin

Full Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-2410

### Affiliations

• Membre Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Tests de permutation d'indépendance en analyse multivariée Theses and supervised dissertations / 2016-11
Guetsop Nangue, Aurélien
Abstract
The main result establishes the equivalence in terms of power between the alpha-distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distribution of index alpha with sufficiently small scale parameters. Large-scale simulations reveal the superiority of these two tests over other tests based on the empirical independence copula process. They also establish the usefulness of the lesser known Pearson type III approximation to the exact permutation distribution. This approximation yields tests with more accurate type I error rates than the gamma approximation usually used for HSIC, especially when dimensions of the two vectors are large. A new method for scale parameter selection in HSIC tests is proposed which improves power performance in three simulations, two of which are from machine learning. The problem of testing mutual independence between many random vectors is addressed. The closely related problem of testing serial independence of a multivariate stationary sequence is also considered. The Möbius transformation of characteristic functions is used to characterize independence. A generalization to p vectors of the alpha -distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distributionof index alpha is obtained. It is shown that an HSIC test with sufficiently small scale parameters is equivalent to an alpha -distance covariance test. Weak convergence of the HSIC test is established. A very fast and accurate computation of p-values uses the Pearson type III approximation which successfully approaches the exact permutation distribution of the tests. This approximation relies on the exact first three moments of the permutation distribution of any test which can be expressed as the sum of all elements of a componentwise product of p doubly-centered matrices. The alpha -distance covariance test and the HSIC test are both of this form. A new selection method is proposed for the scale parameter of the characteristic kernel of the HSIC test. It is shown in a simulation that this adaptive HSIC test has higher power than the alpha-distance covariance test when data are generated from a Student copula. Applications are given to environmental and financial data.

Test d'adéquation à la loi de Poisson bivariée au moyen de la fonction caractéristique Theses and supervised dissertations / 2016-09
Koné, Fangahagnian
Abstract
Our aim in this thesis is to conduct the goodness-of-fit test based on empirical characteristic functions proposed by Jiménez-Gamero et al. (2009) in the case of the bivariate Poisson distribution. We first evaluate the test’s behaviour in the case of the univariate Poisson distribution and find that the estimated type I error probabilities are close to the nominal values. Next, we extend it to the bivariate case and calculate and compare its power with the dispersion index test for the bivariate Poisson, Crockett’s Quick test for the bivariate Poisson and the two test families proposed by Novoa-Muñoz et Jiménez-Gamero (2014). Simulation results show that the probability of type I error is close to the claimed level and that it is generally less powerful than other tests. We also discovered that the dispersion index test should be bilateral whereas it rejects for large values only. Finally, the p-value of all these tests is calculated on a real dataset from soccer. The p-value of the test is 0,009 and we reject the hypothesis that the data come from a Poisson bivariate while the tests proposed by Novoa-Muñoz et Jiménez-Gamero (2014) leads to a different conclusion.

Modèle de mélange de lois multinormales appliqué à l'analyse de comportements et d'habiletés cognitives d'enfants Theses and supervised dissertations / 2011-11
Giguère, Charles-Édouard
Abstract
This study is about the use of mixture to model behavioral and cognitive data measured repeatedly across development in children. Estimation of multinormal mixture models using the EM algorithm is explained in detail. This algorithm simplifies computation of mixture models because the parameters in each group are estimated separately, allowing to model covariance across time more easily. This last point is often disregarded when estimating mixture models. This study focused on the consequences of a misspecified covariance matrix when estimating the number of groups in a mixture. The main consequence is an overestimation of the number of groups, i.e. we estimate groups that do not exist. In particular, the independence assumption of the observations across time when they were in fact correlated resulted in estimating many non existing groups. This overestimation of the number of groups also resulted in an overfit of the model, i.e. we used more parameters than necessary. Finally mixture models were fitted to behavioral and cognitive data. We fitted the data first assuming a covariance structure, then assuming independence. In most cases, the analyses conducted assuming a covariance structure ended up having fewer groups and the results were simpler and clearer to interpret.

Actuarial applications of multivariate phase-type distributions : model calibration and credibility Theses and supervised dissertations / 2009
Abstract
Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

Méthodes de prévision en régression linéaire multivariée Theses and supervised dissertations / 2006
Gueorguieva, Ana
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Moyenne conditionnelle tronquée pour un portefeuille de risques corrélés Theses and supervised dissertations / 2005
Ermilov, Andrey
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Tests d'indépendance en analyse multivariée et tests de normalité dans les modèles ARMA Theses and supervised dissertations / 2002-12
Lafaye de Micheaux, Pierre
Abstract
Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

### Selected publications Expand allCollapse all

#### Graphical lassos for meta-elliptical distributions

Bilodeau, Martin, Graphical lassos for meta-elliptical distributions 42, 185--203 (2014), , Canad. J. Statist.

#### Fitting bivariate losses with phase-type distributions

Bilodeau M, Hassan Zadeh A, Fitting bivariate losses with phase-type distributions , (2011), , Scandinavian Actuarial Journal

#### Multiscale codependence analysis: an integrated approach to analyse relationships accross scales

Guénard G., Legendre P., Boisclair D. and Bilodeau M., Multiscale codependence analysis: an integrated approach to analyse relationships accross scales 91, 2952-2964 (2010), , Ecology

#### A-dependence statistics for mutual and serial independence of categorical variables

Bilodeau M., Lafaye de Micheaux P., A-dependence statistics for mutual and serial independence of categorical variables 139, 2407-2419 (2009), , Journal of Statistical Planning and Inference

#### Nonparametric tests of independence between random vectors

Beran R, Bilodeau M, Lafaye de Micheaux P, Nonparametric tests of independence between random vectors 98, 1805-1824 (2007), , Journal of Multivariate Analysis

#### A multivariate empirical characteristic function test of independence with normal marginals

Bilodeau M. and Lafaye de Micheaux P, A multivariate empirical characteristic function test of independence with normal marginals 95, 345-369 (2005), , Journal of Multivariate Analysis

#### Discussion: Tail conditional expectations for elliptical distributions

Bilodeau M, Discussion: Tail conditional expectations for elliptical distributions 8, 118-123 (2004), , North American Actuarial Journal

#### Asymptotic distribution of the largest eigenvalue

Bilodeau M, Asymptotic distribution of the largest eigenvalue 31, 357-373 (2002), , Communications in Statistics-Simulations and Computations

#### Principal component analysis from multivariate familial correlation matrix

Bilodeau M, Duchesne P, Principal component analysis from multivariate familial correlation matrix 82, 457-470 (2002), , Journal of MUltivariate Analysis

#### Discussion: Robust estimation of the tail index of a Pareto distribution

Bilodeau M, Discussion: Robust estimation of the tail index of a Pareto distribution 5, 123-126 (2001), , North American Actuarial Journal

#### Robust and efficient estimation of the tail index of a single-parameter Pareto distribution'', Vytaras Brazauskas and Robert Serfling, October 2000

Bilodeau, Martin, Robust and efficient estimation of the tail index of a single-parameter Pareto distribution'', Vytaras Brazauskas and Robert Serfling, October 2000 5, 123--128 (2001), , N. Am. Actuar. J.

#### Robust estimation of the SUR Model

Bilodeau M, Duchesne P, Robust estimation of the SUR Model 28, 277-288 (2000), , Canadian Journal of Statistics

#### Multivariate Flattening for better Predictions

Bilodeau M, Multivariate Flattening for better Predictions 28, 159-170 (2000), , Canadian Journal of Statistics

#### Theory of multivariate statistics

Bilodeau M, Brenner D, Theory of multivariate statistics , (1999), , Springer Texts in Statistics, Springer-Verlag (ISBN 0-387-98739-8)

#### Estimating a multivariate treatment effect under a biased allocation rule

Bilodeau M, Estimating a multivariate treatment effect under a biased allocation rule 26, 1119-1124 (1997), , Communications in Statistics-Theory and Methods

#### Some remarks on U(p;m,n) distributions

Bilodeau M, Some remarks on U(p;m,n) distributions 31, 41-43 (1996), , Statistics and Probability Letters

#### Minimax estimators of the mean vector in normal mixed linear models

Bilodeau M, Minimax estimators of the mean vector in normal mixed linear models 52, 73-82 (1995), , Journal of Multivariate Analysis

#### LBI Tests of independence in bivariate exponential distributions

Bilodeau M, Kariya T, LBI Tests of independence in bivariate exponential distributions 46, 127-136 (1994), , Annals of the Institute of Statistical Mathematics

#### Estimation of the eigenvalues of $\Sigma_1\Sigma_2^{-1}$

Bilodeau M, Srivastava M S, Estimation of the eigenvalues of $\Sigma_1\Sigma_2^{-1}$ 41, 1-13 (1992), , Journal of Multivariate Analysis

#### Fourier smoother and generalized additive model

Bilodeau M, Fourier smoother and generalized additive model 20, 257-269 (1992), , Canadian Journal of Statistics

#### Fourier smoother and additive models

Bilodeau, Martin, Fourier smoother and additive models 20, 257--269 (1992), , Canad. J. Statist.

#### On the choice of the loss function in covariance estimation

Bilodeau M, On the choice of the loss function in covariance estimation 8, 131-139 (1990), , Statistics & Decisions

#### On the monotone regression dependence for archimedian bivariate uniform

Bilodeau M, On the monotone regression dependence for archimedian bivariate uniform 18, 981-988 (1989), , Communications in Statistics-Theory and Methods

#### Stein estimation under elliptical distributions

Srivastava M. S., Bilodeau M., Stein estimation under elliptical distributions 28, 247-259 (1989), , Journal of Multivariate Analysis

#### Minimax estimators in the normal MANOVA model

Bilodeau M, Kariya T, Minimax estimators in the normal MANOVA model 28, 260-270 (1989), , Journal of Multivariate Analysis

#### Estimation of the MSE matrix of the Stein estimator

Bilodeau M, Srivastava M S, Estimation of the MSE matrix of the Stein estimator 16, 153-159 (1988), , Canadian Journal of Statistics

#### On the simultaneous estimation of scale-parameters

Bilodeau M, On the simultaneous estimation of scale-parameters 16, 169-174 (1988), , Canadian Journal of Statistics

#### Sur une représentation explicite des solutions optimales d'un programme linéaire

Bilodeau, Martin, Sur une représentation explicite des solutions optimales d'un programme linéaire 29, 419--425 (1986), , Canad. Math. Bull.