The Extent of Gross Errors Eliminated by Robust Multiple Linear Regressions

Abstract

Robust estimation methods can effectively eliminate the influence of gross errors on parameter estimation. However, the extent of gross errors eliminated (EGEE) by robust estimation methods is far-reaching. This article presents a new approach to determine EGEE by robust estimation method. Taking multiple linear regressions (2–5) as examples, simulation experiments were conducted to compare the EGEE of 14 frequently used robust estimation methods. This article confirms several additional efficient robust estimation methods for dealing with multiple linear regressions, as well as the minimum number of observations needed to eliminate gross errors in certain ranges completely.

Keywords:

• Minimum number of observations
• Multiple linear regression
• Robust estimation
• The extents of gross errors eliminated

Mathematics Subject Classification:

• 62G35
• 62J05

Acknowledgments

We would like to thank Prof. N. Balakrishnan, the Editor in Chief, and the two anonymous reviewers for their valued and constructive comments on this article.

Notes

Note. Here, n indicates the number of true errors; β means the factor of minimum mean square true errors; and k indicates lower limit of EGEE.

Note. MSRTE =Mean square residual true error. GE denotes the value of the gross error. M1 = Huber method; M2 = L ₁ method; M3 = L ₁ − L ₂ method; M4 = Andrews method; M5 = Hampel method; M6 = Welsch method; M7 = Tukey method; M8 = Danish method; M9 = Fair method; M10 = Geman-McClure method; M11 = IGG scheme; M12 = IGGIII scheme; M13 = Cauchy method; and M14 = Logistic method. The contents of Table 4 are the same as those of Table 2.

Note. EGEE =Extent of gross errors eliminated. GE denotes the value of the gross error. E1 = Huber method; E2 = L ₁ method; E3 = L ₁ − L ₂ method; E4 = Andrews method; E5 = Hampel method; E6 = Welsch method; E7 = Tukey method; E8 = Danish method; E9 = Fair method; E10 = Geman-McClure method; E11 = IGG scheme; E12 = IGGIII scheme; E13 = Cauchy method; and E14 = Logistic method. The contents of Table 5 are the same as those of Table 3.

Note. MSRTE =Mean square residual true error.

Note. EGEE =Extent of gross errors eliminated.

Note. EGEE =Extent of gross errors eliminated. Here, n indicates the number of observations, whereas g indicates the number of observations that include gross errors. E1 = Huber method; E2 = L ₁ method; E3 = L ₁ − L ₂ method; E4 = Andrews method; E5 = Hampel method; E6 = Welsch method; E7 = Tukey method; E8 = Danish method; E9 = Fair method; E10 = Geman-McClure method; E11 = IGG scheme; E12 = IGGIII scheme; E13 = Cauchy method; and E14 = Logistic method. The contents of Tables 7–9 are the same as those of Table 6.

Note. EGEE =Extent of gross errors eliminated.

Note. EGEE =Extent of gross errors eliminated.

Note. EGEE =Extent of gross errors eliminated.