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Abstract

The well-known Winsor’s principle states that all the distributions are normal in the middle. Two new smoothly redescending ψ-functions based on Winsor’s principle are proposed in the family of M-estimators. The central sections of both of these new ψ-functions resembles with that of the mean, which is linear and this linearity is the actual reason of highest efficiency of the mean under the assumption of normality. The efficiency of an estimator is inversely related to the severity of its robustness. We show that in the class of redescending MEstimators, this new approach produces asymptotically very efficient ψ-functions than that of any other earlier one, while still robust against outliers. The Iteratively Re-weighted Least Squares (IRLS) method based on the proposed ψ-functions clearly detect outliers and ignoring those outliers by giving then zero weights. Two examples selected from the relevant literature, are used for illustrative purposes. The Weighted Least Squares (WLS) method based on the proposed new ψ-functions indeed achieve the goals for which it is constructed. It gives quite improved and satisfactory results in all situations.