Inférence robuste à la présence des valeurs aberrantes dans les enquêtes

This thesis focuses on the treatment of representative outliers in two important aspects of surveys: small area estimation and imputation for item non-response.    Concerning small area estimation, robust estimators in unit-level models have been studied. Sinha & Rao (2009) proposed estimation procedures designed for small area means, based on robustified maximum likelihood parameters estimates of linear mixed model and robust empirical best linear unbiased predictors of the random effect of the underlying model. Their robust methods for estimating area means are of the plug-in type, and in view of the results of Chambers (1986), the resulting robust estimators may be biased in some situations. Biascorrected estimators have been proposed by Chambers et al. (2014). In addition, these robust small area estimators were associated with the estimation of the Mean Square Error (MSE). Sinha & Rao (2009) proposed a parametric bootstrap procedure based on the robust estimates of the parameters of the underlying linear mixed model to estimate the MSE. Analytical procedures for the estimation of the MSE have been proposed in Chambers et al. (2014). However, their theoretical validity has not been formally established and their empirical performances are not fully satisfactorily.  Here, we investigate two new approaches for the robust version the best empirical unbiased estimator: the first one relies on the work of Chambers (1986), while the second proposal uses the concept of conditional bias as an influence measure to assess the impact of units in the population. These two classes of robust small area estimators also include a correction term for the bias. However, they are both fully bias-corrected, in the sense that the correction term takes into account the potential impact of the other domains on the small area of interest unlike the one of Chambers et al. (2014) which focuses only on the domain of interest. Under certain conditions, non-negligible bias is expected for the Sinha-Rao method, while the proposed methods exhibit significant bias reduction, controlled by appropriate choices of the influence function and tuning constants. Monte Carlo simulations are conducted, and comparisons are made between: the new robust estimators, the Sinha-Rao estimator, and the bias-corrected estimator. Empirical results suggest that the Sinha-Rao method and the bias-adjusted estimator of Chambers et al (2014) may exhibit a large bias, while the new procedures offer often better performances in terms of bias and mean squared error.  In addition, we propose a new bootstrap procedure for MSE estimation of robust small area predictors. Unlike existing approaches, we formally prove the asymptotic validity of the proposed bootstrap method. Moreover, the proposed method is semi-parametric, i.e., it does not rely on specific distributional assumptions about the errors and random effects of the unit-level model underlying the small-area estimation, thus it is particularly attractive and more widely applicable. We assess the finite sample performance of our bootstrap estimator through Monte Carlo simulations. The results show that our procedure performs satisfactorily well and outperforms existing ones. Application of the proposed method is illustrated by analyzing a well-known outlier-contaminated small county crops area data from North-Central Iowa farms and Landsat satellite images.   Concerning imputation in the presence of item non-response, some single imputation methods have been studied. The deterministic regression imputation, which includes the ratio imputation and mean imputation are often used in surveys. These imputation methods may lead to biased imputed estimators if the imputation model or the non-response model is not properly specified. Recently, doubly robust imputed estimators have been developed. However, in the presence of outliers, the doubly robust imputed estimators can be very unstable. Using the concept of conditional bias as a measure of influence (Beaumont, Haziza and Ruiz-Gazen, 2013), we propose an outlier robust version of the doubly robust imputed estimator. Thus this estimator is denoted as a triple robust imputed estimator. The results of simulation studies show that the proposed estimator performs satisfactorily well for an appropriate choice of the tuning constant.

Keyphrases: Bootstrap, Imputation, Corrected-bias estimator, Conditional bias, Outliers, Model-based inference, Sampling-based inference, Small-area, Linear mixed model, Robustness