Department of Economics and Business Economics

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

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  • Soren Johansen
  • Bent Nielsen, University of Oxford - Nuffield College, United Kingdom

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

Original languageEnglish
Pages (from-to)1131-1183
Number of pages53
Publication statusPublished - 1 May 2016

    Research areas

  • Fixed point result, Forward Search, Iterated exponential martingale inequality, Quantile process, Weighted and marked empirical process

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