Open Access
May 2016 Analysis of the Forward Search using some new results for martingales and empirical processes
Søren Johansen, Bent Nielsen
Bernoulli 22(2): 1131-1183 (May 2016). DOI: 10.3150/14-BEJ689


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.


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Søren Johansen. Bent Nielsen. "Analysis of the Forward Search using some new results for martingales and empirical processes." Bernoulli 22 (2) 1131 - 1183, May 2016.


Received: 1 February 2013; Revised: 1 November 2014; Published: May 2016
First available in Project Euclid: 9 November 2015

zbMATH: 06562308
MathSciNet: MR3449811
Digital Object Identifier: 10.3150/14-BEJ689

Keywords: fixed point result , Forward search , iterated exponential martingale inequality , quantile process , weighted and marked empirical process

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 2 • May 2016
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