Statistical Consultants Ltd
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:
The ranks are used to calculate weights. For example:
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, must equal zero. Several rules for ties can be used to keep this expression true. They include:
Jaeckel’s estimatorThis was proposed by Jaeckel in 1972. It involves finding the beta coefficient that minimises D:
He found that D is non-negative and beta is asymptotically normal with mean β.
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