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Contributions to the theory of robust inference Salibian-Barrera, Matias

Abstract

We study the problem of performing statistical inference based on robust estimates when the distribution of the data is only assumed to belong to a contamination neighbourhood of a known central distribution. We start by determining the asymptotic properties of some robust estimates when the data are not generated by the central distribution of the contamination neighbourhood. Under certain regularity conditions the considered estimates are consistent and asymptotically normal. For the location model and with additional regularity conditions we show that the convergence is uniform on the contamination neighbourhood. We determine that a class of robust estimates satisfies these requirements for certain proportions of contamination, and that there is a trade-off between the robustness of the estimates and the extent to which the uniformity of their asymptotic properties holds. When the distribution of the data is not the central distribution of the neighbourhood the asymptotic variance of these estimates is involved and difficult to estimate. This problem affects the performance of inference methods based on the empirical estimates of the asymptotic variance. We present a new re-sampling method based on Efron's bootstrap (Efron, 1979) to estimate the sampling distribution of MM-location and regression estimates. This method overcomes the main drawbacks of the use of bootstrap with robust estimates on large and potentially contaminated data sets. We show that our proposal is computationally simple and that it provides stable estimates when the data contain outliers. This new method extends naturally to the linear regression model.

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