La classe des M-estimateurs engendre des estimateurs classiques d’un paramètre de localisation multidimensionnel tels que l’estimateur du maximum de vraisemblance, la moyenne empirique et la médiane spatiale. Huber (1964) introduit les M-estimateurs dans le cadre de l’étude des estimateurs robustes. Parmi la littérature dédiée à ces estimateurs, on trouve en particulier les ouvrages de Huber (1981) et de Hampel et al. (1986) sur le comportement asymptotique et la robustesse via le point de rupture et la fonction d’influence (voir Ruiz-Gazen (2012) pour une synthèse sur ces notions). Plus récemment, des résultats sur la convergence et la normalité asymptotique sont établis par Van der Vaart (2000) dans le cadre multidimensionnel. Nevalainen et al. (2006, 2007) étudient le cas particulier de la médiane spatiale pondérée et non-pondérée dans le cas clusterisé. Nous généralisons ces résultats aux M-estimateurs pondérés. Nous étudions leur convergence presque sûre, leur normalité asymptotique ainsi que leur robustesse dans le cas de données clusterisées. / M-estimators were first introduced by Huber (1964) as robust estimators of location and gave rise to a substantial literature. For results on their asymptotic behavior and robustness (using the study of the influence func- tion and the breakdown point), we may refer in particular to the books of Huber (1981) and Hampel et al. (1986). For more recent references, we may cite the work of Ruiz-Gazen (2012) with a nice introductory presentation of robust statistics, and the book of Van der Vaart (2000) for results, in the independent and identically distributed setting, concerning convergence and asymptotic normality in the multivariate setting considered throughout this paper. Most of references address the case where the data are independent and identically distributed. However clustered, and hierarchical, data frequently arise in applications. Typically the facility location problem is an important research topic in spatial data analysis for the geographic location of some economic activity. In this field, recent studies perform spatial modelling with clustered data (see e.g. Liao and Guo, 2008; Javadi and Shahrabi, 2014, and references therein). Concerning robust estimation, Nevalainen et al. (2006) study the spatial median for the multivariate one-sample location problem with clustered data. They show that the intra-cluster correlation has an impact on the asymptotic covariance matrix. The weighted spatial median, introduced in their pioneer paper of 2007, has a superior efficiency with respect to its unweighted version, especially when clusters’ sizes are heterogenous or in the presence of strong intra-cluster correlation. The class of weighted M-estimators (introduced in El Asri, 2013) may be viewed as a generalization of this work to a broad class of estimators: weights are assigned to the objective function that defines M-estimators. The aim is, for example, to adapt M-estimators to the clustered structures, to the size of clusters, or to clusters including extremal values, in order to increase their efficiency or robustness. In this thesis, we study the almost sure convergence of unweighted and weighted M-estimators and establish their asymptotic normality. Then, we provide consistent estimators of the asymptotic variance and derived, numerically, optimal weights that improve the relative efficiency to their unweighted versions. Finally, from a weight-based formulation of the breakdown point, we illustrate how these optimal weights lead to an altered breakdown point.
| Identifer | oai:union.ndltd.org:theses.fr/2014AVIG0411 |
| Date | 15 December 2014 |
| Creators | El Asri, Mohamed |
| Contributors | Avignon, Blanke, Delphine, Gabriel, Edith |
| Source Sets | Dépôt national des thèses électroniques françaises |
| Language | French |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
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