Robust estimation based on a novel family of arctan disparities and the limitation of the second order influence function

The article presents a new family of disparity measures based on the trigonometric tan-1 function. A subset of members from the proposed disparity family are shown to have excellent robustness properties against both inliers and outliers and are competitive with other popular disparities such as the Hellinger distance and the symmetric chi-square. The most notable thing about the presented disparity is that the strong robustness properties of the corresponding minimum distance estimators are attained in spite of predictions to the contrary by not just the first order influence function but also the second order influence function. This demonstrates the limitation of even the second order influence analysis in predicting the robustness properties of a disparity. Several examples and numerical studies illustrate the aforementioned property. Copyright (c) 2018 John Wiley & Sons, Ltd.