SENSITIVITY ANALYSIS OF M-ESTIMATES

JAN ÁMOS VÍŠEK

Department of Stochastic Informatics, Institute of Information Theory and Automation,

Academy of Sciences the Czech Republic, and Department of Macroeconomics, Institute of Economic Studies, Charles University, Czech Republic

(Received March 22, 1994; revised June 26, 1995)

Abstract. Bahadur representation of the difference of estimators of regression coefficients for the full data set and for the set from which one observation was deleted is given for the M-estimators which are generated by a continuous ψ-function. The representation is invariant with respect to the scale of residuals and it indicates that the bound of the norm of the difference is proportional to the gross error sensitivity. Then for the ψ-function which corresponds to the median it is shown that the difference of the estimates for the full data and for data without one observation, although being bounded in probability, can be much larger than indicated by the gross error sensitivity.

Key words and phrases: Sensitivity analysis, influential points in M-esti-mation, scale invariance.