Discussion: The forward search: Theory and data analysis

Abstract

In the last decade the forward search has emerged not only as an algorithm for finding robust estimates, but also as a method for providing a broad range of exploratory (and confirmatory) procedures for analyzing data, such as data transformation and variable selection. For its original purpose of being a “fast very robust” procedure (Atkinson, 1994), the forward search cannot compete with the “fast algorithm” of Rousseeuw and van Driessen (1999, 2006) in terms of velocity. The basic idea behind the fast algorithm consists of carrying out many two-step procedures: a trial step followed by a refinement step (the so-called Concentration step). The latter step especially shows a “fast” (convergence) property. The authors are to be congratulated on the achievement of filling in the theoretical part for the forward search. The forward search also has several advantages for diagnostic plotting, such as the stalactite plot (Atkinson & Mulira, 1993), fan plot (Atkinson & Riani, 2000), and candlestick plot (Atkinson, Riani, & Cerioli, 2010). These contribute insights into the data analysis that no other available algorithms for robust estimators can. We present herein another advantage of the forward search this time for the heteroscedastic regression model (Cheng, submitted for publication). The benefit is from the nature of the forward search algorithm — the number of observations is augmented in a way that not only excludes the potentially outlying cases, but also takes into account the heteroscedastic structure of the data.