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

Søren Johansen; Bent Nielsen

doi:10.3150/14-bej689

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

13 Citationer (Scopus)

Abstract

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.

Originalsprog	Engelsk
Tidsskrift	Bernoulli
Vol/bind	22
Udgave nummer	2
Sider (fra-til)	1131-1183
Antal sider	53
ISSN	1350-7265
DOI	https://doi.org/10.3150/14-bej689
Status	Udgivet - maj 2016

Adgang til dokumentet

10.3150/14-bej689

Analysis_of_the_Forward_Search_using_some_new_results_for_martingales_and_empirical_processes_VOR16.pdfForlagets udgivne version, 573 KB

Citationsformater

@article{63cc41ddae4f476eafe6a7940e938c02,

title = "Analysis of the Forward Search using some new results for martingales and empirical processes",

abstract = "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.",

author = "S{\o}ren Johansen and Bent Nielsen",

year = "2016",

month = may,

doi = "10.3150/14-bej689",

language = "English",

volume = "22",

pages = "1131--1183",

journal = "Bernoulli",

issn = "1350-7265",

publisher = "International Statistical Institute",

number = "2",

}

TY - JOUR

T1 - Analysis of the Forward Search using some new results for martingales and empirical processes

AU - Johansen, Søren

AU - Nielsen, Bent

PY - 2016/5

Y1 - 2016/5

N2 - 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.

AB - 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.

U2 - 10.3150/14-bej689

DO - 10.3150/14-bej689

M3 - Journal article

SN - 1350-7265

VL - 22

SP - 1131

EP - 1183

JO - Bernoulli

JF - Bernoulli

IS - 2

ER -

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

Abstract

Adgang til dokumentet

Fingeraftryk

Citationsformater