Discussiones Mathematicae Probability and Statistics 20(1) (2000) 25-50

Brenton R. Clarke

*Mathematics and Statistics, Division of Science and Engineering
Murdoch University
Murdoch, W.A., 6150, Australia *

In small to moderate sample sizes it is important to make use of all the data when there are no outliers, for reasons of efficiency. It is equally important to guard against the possibility that there may be single or multiple outliers which can have disastrous effects on normal theory least squares estimation and inference. The purpose of this paper is to describe and illustrate the use of an adaptive regression estimation algorithm which can be used to highlight outliers, either single or multiple of varying number. The outliers can include 'bad' leverage points. Illustration is given of how 'good' leverage points are retained and 'bad' leverage points discarded. The adaptive regression estimator generalizes its high breakdown point adaptive location estimator counterpart and thus is expected to have high efficiency at the normal model. Simulations confirm this. On the other hand, examples demonstrate that the regression algorithm given highlights outliers and 'potential' outliers for closer scrutiny.

The algorithm is computer intensive for the reason that it is a global algorithm which is designed to highlight outliers automatically. This also obviates the problem of searching out ``local minima" encountered by some algorithms designed as fast search methods. Instead the objective here is to assess all observations and subsets of observations with the intention of culling all outliers which can range up to as much as approximately half the data. It is assumed that the distributional form of the data less outliers is approximately normal. If this distributional assumption fails, plots can be used to indicate such failure, and, transformations may be ;required before potential outliers are deemed as outliers. A well known set of data illustrates this point.

**Keywords:** outlier; least median of squares regression; least trimmed squares;
trimmed likelihood; adaptive estimation; leverage.

**1991 Mathematics Subject Classification:** 62T05.

[1] | A.C. Atkinson, Two graphical displays for outlying and influential observations in
regression, Biometrika 68 (1981), 13-20. |

[2] | A.C. Atkinson, Masking unmasked, Biometrika 73 (1986a), 533-41. |

[3] | A.C. Atkinson, Comment : Aspects of diagnostic regression analysis,
Statistical Science 1 (1986b), 397-401. |

[4] | A.C. Atkinson, Fast very robust methods for the detection of multiple outliers,
Journal of the American Statistical Association 89 (1994), 1329-1339. |

[5] | V. Barnett and T. Lewis, Outliers in Statistical Data, 3rd ed., New York, Wiley, 1994. |

[6] | T. Bednarski and B.R. Clarke, Trimmed likelihood estimation of location and scale
of the normal distribution, Australian Journal of Statistics 35 (1993),
141-153. |

[7] | M.D. Brown, J. Durbin and J.M. Evans, Techniques for testing the constancy of
regression relationships over time, Journal of the Royal Statistical Society, Series B
37 (1975), 149-192. |

[8] | K.A. Brownlee, Statistical Theory and Methodology in Science and Engineering, 2^{nd}
ed., New York, Wiley, 1965. |

[9] | R.W. Butler, Nonparametric interval and point prediction using data trimmed by a
Grubbs-type outlier rule, Annals of Statistics 10 (1982), 197-204. |

[10] | R.L. Chambers and C.R. Heathcote, On the estimation of slope and the
identification of outliers in linear regression, Biometrika 68 (1981), 21-33. |

[11] | B.R. Clarke, Empirical evidence for adaptive confidence intervals and
identification of outliers using methods of trimming, Australian Journal of Statistics
36 (1994), 45-58. |

[12] | R.D. Cook and S. Weisberg, Residuals and Influence in Regression, New York and London, Chapman and Hall 1982. |

[13] | P.L. Davies, The asymptotics of S-estimators in the Linear Regression Model,
Annals of Statistics 18 (1990), 1651-1675. |

[14] | P.L. Davies and U. Gather, The identification of multiple outliers (with
discussion), Journal of the American Statistical Association 88 (1993),
782-801. |

[15] | N.R. Draper and H. Smith, Applied Regression Analysis, New York, Wiley, 1966. |

[16] | W. Fung, Unmasking outliers and leverage points : A confirmation, Journal of
the American Statistical Association 88 (1993), 515-519. |

[17] | A.S. Hadi and J.S. Simonoff, Procedures for the identification of multiple
outliers in linear models, Journal of the American Statistical Association 88
(1993), 1264-1272. |

[18] | F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.J. Stahel, Robust Statistics, the Approach Based on Influence Functions, New York, Wiley, 1986. |

[19] | D.M. Hawkins, D. Bradu and G.V. Kass, Location of several outliers in
multiple-regression data using elemental sets, Technometrics 26 (1984),
197-208. |

[20] | T.P. Hettmansperger and S.J. Sheather, A cautionary note on the method of least
median squares, American Statistician 46 (1992), 79-83. |

[21] | L.A. Jaeckel, Some flexible estimates of location, Annals of Mathematical
Statistics 42 (1971), 1540-1552. |

[22] | F. Kianifard and W.H. Swallow, Using recursive residuals, calculated on
adaptively-ordered observations, to identify outliers in linear regression, Biometrics
45 (1989), 571-585. |

[23] | F. Kianifard and W.H. Swallow, A Monte Carlo comparison of five procedures for
identifying outliers in linear regression, Communications in Statistics, Part A-Theory
and Methods 19 (1990), 1913-1938. |

[24] | M.G Marasinghe, A multistage procedure for detecting several outliers in linear
regression, Technometrics 27 (1985), 395-399. |

[25] | P.J. Rousseeuw, Least median of squares regression, Journal of the American
Statistical Association 79 (1984), 871-880. |

[26] | P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection, New York, Wiley, 1987. |

[27] | P.J. Rousseeuw and B.C. van Zomeren, Unmasking multivariate outliers and leverage
points, Journal of the American Statistical Association 85 (1990), 633-651. |

[28] | P.J. Rousseeuw and V.J. Yohai, Robust regression by means of S-estimators,
in: Robust and Nonlinear Time Series Analysis, eds., J. Franke, W. Härdle and
R.D. Martin, (Lecture Notes in Statistics), New York, Springer-Verlag, (1984), 256-272. |

[29] | D. Ruppert, Computing S-estimators for regression and multivariate
location/dispersion, Journal of Computational and Graphical Statistics 1
(1992), 253-270. |

[30] | T.P. Ryan, Comment on Hadi and Simonoff, Letters to the Editor, Journal of the
American Statistical Association 90 (1995), 811. |

[31] | G. Simpson, D. Ruppert and R.J. Carroll, On one-step GM estimates and stability
of inferences in linear regression, Journal of the American Statistical Association 87
(1992), 439-450. |

[32] | W.H. Swallow and F. Kianifard, Using robust scale estimates in detecting multiple
outliers in linear regression, Biometrics 52 (1996), 545-556. |

[33] | J.W Tukey and D.H. McLaughlin, Less vulnerable confidence and significance
procedures for location based on a single sample: Trimming/Winsorization 1, Sankhya 25
(A) (1963), 331-352. |

[34] | W.N. Venables and B.D. Ripley, Modern Applied Statistics with S-Plus, New York, Springer-Verlag, 1994. |

[35] | D.L. Woodruff and D.M. Rocke, Computable robust estimation of multivariate
location and shape in high dimension using compound estimators, Journal of the
American Statistical Association 89 (1994), 888-896. |

[36] | V.J. Yohai, High breakdown point and high-efficiency robust estimates for
regression, Annals of Statistics 15 (1987), 642-656. |

[37] | V.J. Yohai and R.H. Zamar, High breakdown-point estimates of regression by means
of the minimization of an efficient scale, Journal of the American Statistical
Association 83 (1988), 406-413. |

Received 15 November 1998