Corrigendum: Analysis of the forward search using some new results for martingales and empirical processes

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.

[1]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[2]  Bent Nielsen,et al.  Discussion of The Forward Search: Theory and Data Analysis , 2010 .

[3]  Bent Nielsen,et al.  An Analysis of the Indicator Saturation Estimator as a Robust Regression Estimator , 2008 .

[4]  Jan Ámos Vísek,et al.  The least trimmed squares. Part III: Asymptotic normality , 2006, Kybernetika.

[5]  David J. Olive,et al.  Inconsistency of Resampling Algorithms for High-Breakdown Regression Estimators and a New Algorithm , 2002 .

[6]  M. Csorgo.  Quantile processes with statistical applications , 1987 .

[7]  A. Hadi.  Identifying Multiple Outliers in Multivariate Data , 1992 .

[8]  A. Atkinson,et al.  Finding an unknown number of multivariate outliers , 2009 .

[9]  Robert G. Staudte,et al.  Influence Functions of Iteratively Reweighted Least Squares Estimators , 1991 .

[10]  Anthony C. Atkinson,et al.  Robust Diagnostic Regression Analysis , 2000 .

[11]  D. G. Simpson,et al.  On One-Step GM Estimates and Stability of Inferences in Linear Regression , 1992 .

[12]  R. R. Bahadur.  A Note on Quantiles in Large Samples , 1966 .

[13]  D. Ruppert,et al.  Trimmed Least Squares Estimation in the Linear Model , 1980 .

[14]  Sangyeol Lee,et al.  On residual empirical processes of stochastic regression models with applications to time series , 1999 .

[15]  Jan Ámos Vísek,et al.  The least trimmed squares. Part I: Consistency , 2006, Kybernetika.

[16]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[17]  A. Atkinson.  Fast Very Robust Methods for the Detection of Multiple Outliers , 1994 .

[18]  J. Simonoff,et al.  Procedures for the Identification of Multiple Outliers in Linear Models , 1993 .

[19]  Tiziano Bellini,et al.  The forward search interactive outlier detection in cointegrated VAR analysis , 2016, Adv. Data Anal. Classif..

[20]  Alessio Farcomeni,et al.  Strong consistency and robustness of the Forward Search estimator of multivariate location and scatter , 2014, J. Multivar. Anal..

[21]  P. Rousseeuw.  Least Median of Squares Regression , 1984 .

[22]  Bent Nielsen,et al.  Asymptotic Theory of Outlier Detection Algorithms for Linear Time Series Regression Models , 2016 .

[23]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[24]  J. Kiefer.  On Bahadur's Representation of Sample Quantiles , 1967 .

[25]  Leo A. Aroian,et al.  A Study of R. A. Fisher's $z$ Distribution and the Related F Distribution , 1941 .

[26]  E. Ronchetti,et al.  A journey in single steps: robust one-step M-estimation in linear regression , 2002 .

[27]  Inge S. Helland,et al.  Central Limit Theorems for Martingales with Discrete or Continuous Time , 1982 .

[28]  Jan Ámos Vísek,et al.  The least trimmed squares. Part II: \sqrt{n}-consistency , 2006, Kybernetika.

[29]  G. Shorack.  Weak convergence of empirical and quantile processes in sup-norm metrics via kmt-constructions , 1979 .

[30]  Anthony C. Atkinson,et al.  The forward search: theory and data analysis , 2010 .

[31]  H. Koul,et al.  Weak Convergence of Randomly Weighted Dependent Residual Empiricals with Applications to Autoregression , 1994 .

[32]  Giuseppe Cavaliere,et al.  EXPLOITING INFINITE VARIANCE THROUGH DUMMY VARIABLES IN NONSTATIONARY AUTOREGRESSIONS , 2013, Econometric Theory.

[33]  B. Nielsen,et al.  The Empirical Process of Autoregressive Residuals , 2007 .

[34]  Anthony C. Atkinson,et al.  Fast calibrations of the forward search for testing multiple outliers in regression , 2007, Adv. Data Anal. Classif..

[35]  P. Bickel.  One-Step Huber Estimates in the Linear Model , 1975 .

[36]  M. Sampford.  Some Inequalities on Mill's Ratio and Related Functions , 1953 .

[37]  Andrew P. Soms,et al.  An Asymptotic Expansion for the Tail Area of the t -Distribution , 1976 .

[38]  Jan Ámos Víšek,et al.  Least trimmed squares , 2000 .

[39]  E. Ronchetti.  Discussion: The forward search: Theory and data analysis , 2010 .

[40]  Bent Nielsen,et al.  Outlier Detection Algorithms for Least Squares Time Series Regression , 2014 .

[41]  Marco Riani,et al.  Distribution Theory and Simulations for Tests of Outliers in Regression , 2006 .

[42]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[43]  S. Johansen,et al.  Outlier detection in regression using an iterated one-step approximation to the huber-skip estimator , 2013 .

[44]  Bernard Bercu,et al.  Exponential inequalities for self-normalized martingales with applications , 2007, 0707.3715.

[45]  William C. Guenther,et al.  An Easy Method for Obtaining Percentage Points of Order Statistics , 1977 .