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The Forward Search is an iterative algorithm for avoiding outliers in aregression analysis suggested by Hadi and Simonoff (J. Amer. Statist. Assoc. 88(1993) 1264-1272), see also Atkinson and Riani (Robust Diagnostic RegressionAnalysis (2000) Springer). The algorithm constructs subsets of "good"observations so that the size of the subsets increases as the algorithmprogresses. It results in a sequence of regression estimators and forwardresiduals. Outliers are detected by monitoring the sequence of forwardresiduals. We show that the sequences of regression estimators and forwardresiduals converge to Gaussian processes. The proof involves a new iteratedmartingale inequality, a theory for a new class of weighted and markedempirical processes, the corresponding quantile process theory, and a fixedpoint argument to describe the iterative aspect of the procedure.

Analysis of the Forward Search using some new results for martingales and empirical processes