Robust bootstrap regression testing in the presence of outliers

Bootstrap is one of the random sampling methods with replacement, that was proposed to address the problem of small samples whose distributions are difficult to derive.  The distribution of bootstrap samples is empirical or free and due to its random sampling with replacement, the probability of choosing a specific observation may be equal to one.  Unfortunately, when the original sample data contains an outlier, there is a serious problem that leads to a breakdown OLS (Ordinary Least Squares) estimator, and robust regression methods should be recommended.  It is well known that the best robust regression method has a high breakdown point is not more than 0.50, so the robust regression method would break down when the percentage of outliers in the bootstrap sample exceeds 0.50. It is well known that fixed-x bootstrap is resampled the residuals which probably are having outliers.  Moreover, the leverage point(s) is an outlier that occurs in X-direction, so the effects of it on fixed-x bootstrap samples would be existence.  However, the decision-making about the null hypothesis of bootstrap regression coefficients could not be reliable.  In this paper, we propose using weighted fixed-x bootstrap with a probability approach to guarantee the percentage of outliers in the bootstrap samples will be very low.  And then weighted M-estimate should be to tackle the problem of outliers and leverage points and taking a more reliable decision about bootstrap regression coefficients hypothesis test.  The performance of the suggested method has been tested with others by using real data and simulation.  The results show our proposed method is more efficient and reliable than the others.

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