Does anyone know how we can robustly standardize the residuals in MM regression? First we perform MM regression and then obtain the residuals: how can we robustly standardize the residuals obtained from MM regression? I have found the method for least median squares (LMS) and least trimmed mean squares (LTS) in which the scale of the errors is estimated using a formula and then the residuals will be divided by that estimated scale. But for MM regression I could not find a formula for estimating the scale of the errors in order to standardize the residuals.
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The robust scale is normally output by the routine you used to estimate the MM. For example, in R:
library(robustbase)
data(coleman)
set.seed(0)
RlmST <- lmrob(log.light ~ log.Te, data = starsCYG)
RlmST$scale
There is no explicit formula to compute it: it's the result of an iterative scheme.
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$\begingroup$ Thanks for your answer. you are right that its the result if iterative scheme. what I am going to do is to labele the outliers using the standardized resiudals obtained from MM regression. I think if I standardize the residuals as "residuals/sd(residuals)" and then check for large values to be labeled as outliers, this way cant be a good way . And even using RlmST$scale and standardizing as residuals/ $\endgroup$– user22Commented Nov 9, 2013 at 15:59
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$\begingroup$ I want to robustly standardize the residuals obtained using MM regression. For example for LTS(Robust statistics for outlier detection, Rousseeuw et al.), the standardized LTS residuals are obtained by residual/S where S^2=c^2*sum((r_(i))^2)/h..r(i) are ordered residuals. Now y question is that how I can robustly standardize the residuals obtained by using MM regression $\endgroup$– user22Commented Nov 9, 2013 at 16:34
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$\begingroup$ Yes, but only using the "scale" component returned by the MM routine you use. Using the usual s.d. defeats the whole purpose of robust estimation, since it is itself liable to being swindled by the outliers. Also, the whole point of MM estimation was to dispose with outlier identification in the first place. If you're out to identify outliers, you might prefer an estimation approach based on 1-0 weights (like the FLTS). $\endgroup$– user603Commented Nov 9, 2013 at 16:57
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$\begingroup$ I want to use an estimation approach like what you said based on 1-0 weights. Also I tried what you said to standardize the residuals using the scale obtained from MM regression as Sres=res/scale. And then compared the values of the AbsSR=abs(Sres) with 2.5 to weigh the observations. but the results was strange and the values for AbsSR were really large. $\endgroup$– user22Commented Nov 9, 2013 at 17:17
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$\begingroup$ 'for AbsSR were really large' that's okay: it just mean you have large outliers. Presumably, the MM coefficients will be very different from the usual OLS ones. "I want to use an estimation approach like what you said based on 1-0 weights" then use FastLTS. $\endgroup$– user603Commented Nov 10, 2013 at 12:34