Procedures for the Identification of Multiple Outliers in Linear Models

Shift outliers in linear inference

“…Then the beta values for (Y ω −ω 12 ) in column 6 are seen to be  β(Y ω −ω 12 ) =  β ω −β o , generating the discrepancies between  β ω and β o . In particular, these values give Table 1 data of Hadi and Simonoff [10], but that these shifts are accompanied by riveting standardized changes in the betas.…”

“…Accordingly, the angle θ 1 between (ω, ω 1 ) is θ 1 = 50.8°, indicating that ω has only a slight propensity towards the ''Error'' space. Hadi and Simonoff [10] and others offer often intricate subset deletion algorithms for identifying outlying subsets. Our work supports the view that shifts in responses are tantamount to shifts in the OLS solutions, and to inflated variation about the best-fitting line, these being the principal deleterious consequences of outlying data.…”

“…For completeness it should be recorded that the analysis of Hadi and Simonoff [10] in regard to their Table 1 is strictly inadmissible. Specifically, they applied OLS when their first three observations are deterministic since devoid of random disturbances.…”

“… as the bias owing to the shift {Y 1 + 4}, given also as κ in Table 1. Responses for analyses, to include Y ∅ = f (X) + ε, Y ω = (Y ∅ + ω), (Y ∅ + ω 11 ), (Y ω + ω 11 ), and (Y ω − ω 12 ), with ω 11 = X (  β ω − β o ) and ω 12 = X β o ; to include OLS solutions and S 2 for each; partial elements of residuals e ∅ and e ω ; and the bias B(S 2 ) for S 2 taking values S 2 Hadi and Simonoff [10] presented an artificial data set with two predictor variables {X 1 , X 2 } having response Y = X 1 + X 2 + ε, sample size n = 25, and outliers embedded in rows {1, 2, 3} designed to be difficult to find. Their errors were generated from N(0, 1) for rows {4, .…”

The forward search: Theory and data analysis

“…The idea of fitting a model to subsets of an increasing size was introduced by Hadi (1992) and Atkinson (1994) for multivariate data and by Hadi and Simonoff (1993) for regression. The ensuing development of the forward search, combining flexible downweighting with a focus on data analysis, is covered in two books, Atkinson and Riani (2000) on regression and Atkinson, Riani, and Cerioli (2004), which is concerned with multivariate procedures.…”