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 Robust Regression Examples

# Examples Combining Robust Residuals and Robust Distances

This section is based entirely on Rousseeuw and Van Zomeren (1990). Observations xi, which are far away from most of the other observations, are called leverage points. One classical method inspects the Mahalanobis distances MDi to find outliers xi:

where C is the classical sample covariance matrix. Note that the MVE subroutine prints the classical Mahalanobis distances MDi together with the robust distances RDi. In classical linear regression, the diagonal elements hii of the hat matrix
H = X(XTX)-1XT
are used to identify leverage points. Rousseeuw and Van Zomeren (1990) report the following monotone relationship between the hii and MDi
hii = [((MDi)2)/(N-1)] + [1/n]
and point out that neither the MDi nor the hii are entirely safe for detecting leverage points reliably. Multiple outliers do not necessarily have large MDi values because of the masking effect.

The definition of a leverage point is, therefore, based entirely on the outlyingness of xi and is not related to the response value yi. By including the yi value in the definition, Rousseeuw and Van Zomeren (1990) distinguish between the following:

• Good leverage points are points (xi,yi) that are close to the regression plane; that is, good leverage points improve the precision of the regression coefficients.
• Bad leverage points are points (xi,yi) that are far from the regression plane; that is, bad leverage points reduce the precision of the regression coefficients.

Rousseeuw and Van Zomeren (1990) propose to plot the standardized residuals of robust regression (LMS or LTS) versus the robust distances RDi obtained from MVE. Two horizontal lines corresponding to residual values of +2.5 and -2.5 are useful to distinguish between small and large residuals, and one vertical line corresponding to the is used to distinguish between small and large distances.