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R-estimators


Statistical Analysis Techniques, Robust Estimators, Alternatives to OLS


The three main classes of robust estimators are M, L and R.  Robust estimators are resistant to outliers and when used in regression modelling, are robust to departures from the normality assumption.

R-estimators involve ranking residuals.  The rank of a sample is a mapping from n real numbers to the integers 1 through to n.  The smallest number is given rank 1, the next smallest is given rank 2 etc.  For example:

ranking residuals

The ranks are used to calculate weights.  For example:

R estimation

When generating an R vector, ties need to be taken into account.  A tie is when two values getting ranked are equal.  For the weights to work, R estimation rule must equal zero.  Several rules for ties can be used to keep this expression true.  They include:

tied rankings

Jaeckel’s estimator

This was proposed by Jaeckel in 1972.  It involves finding the beta coefficient that minimises D:

Jaeckel's estimator

He found that D is non-negative and beta is asymptotically normal with mean β.


See also:

Regression
M-estimators
L-estimators

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