Robust Bootstrap with Non Random Weights Based on the Influence Function

Abstract The existence of outliers in a sample is an obvious problem which can become worse when the usual bootstrap is applied, because some resamples may have a higher contamination level than the initial sample. Bootstrapping using robust estimators may be a solution to this problem. However, in many instances, this will not be enough because it can lead to several complications, such as: (i) the breakdown point for the whole procedure may be small even when based on an estimator with a high breakdown point; (ii) mathematical difficulties; (iii) very high computation time. In this paper, we suggest a modification of the bootstrap procedure in order to solve these problems which consists of forming each bootstrap sample by resampling with different probabilities so that the potentially more harmful observations have smaller probabilities of selection. The aim is to protect the whole procedure against a given number of arbitrary outliers. As an illustration, we consider point and interval estimation for the correlation coefficient. We use Monte Carlo methods to compare this method with another robust bootstrap procedure, the winsorized bootstrap, recently suggested by Singh [Singh, K. (1998). Breakdown theory for bootstrap quantiles. Ann. Statist. 26:1719–1732].

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