The influence function and maximum bias of Tukey's median

Tukey's median is among one of the earliest known high breakdown point multivariate location statistics. Aside from its breakdown point, though, little else appears to be known about its robustness properties. In this paper we investigate other aspects of Tukey's median, and in particular we derive and study its influence function and its maximum contamination bias function. When judged by these other robustness criteria, Tukey's median again proves to be highly robust.

[1]  K. Mardia.  Statistics of Directional Data , 1972 .

[2]  J. Tukey.  Mathematics and the Picturing of Data , 1975 .

[3]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[4]  G. S. Watson.  Statistics on Spheres , 1983 .

[5]  Christopher G. Small.  Measures of centrality for multivariate and directional distributions , 1987 .

[6]  V. Yohai,et al.  Min-Max Bias Robust Regression. , 1989 .

[7]  H. P. Lopuhaä.  On the relation between S-estimators and M-estimators of multivariate location and covariance , 1989 .

[8]  C. Small.  A Survey of Multidimensional Medians , 1990 .

[9]  D. Donoho,et al.  Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness , 1992 .

[10]  Hendrik P. Lopuhaä,et al.  Highly efficient estimators of multivariate location with high breakdown point , 1992 .

[11]  D. G. Simpson,et al.  Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation , 1993 .

[12]  Zhiqiang Chen.  Robustness of the half-space median , 1995 .

[13]  David E. Tyler,et al.  Constrained M-estimation for multivariate location and scatter , 1996 .

[14]  Z. Bai,et al.  Asymptotic distributions of the maximal depth estimators for regression and multivariate location , 1999 .

[15]  M. Bilodeau,et al.  Theory of multivariate statistics , 1999 .

[16]  On min{max majority and deepest points , 1999 .

[17]  P. Rousseeuw,et al.  The depth function of a population distribution , 1999, Metrika.

[18]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[19]  Ronald H. Randles,et al.  A Simpler, Affine-Invariant, Multivariate, Distribution-Free Sign Test , 2000 .

[20]  V. Yohai,et al.  Projection estimates of multivariate location , 2002 .

[21]  Christophe Croux,et al.  Location adjustment for the minimum volume ellipsoid estimator , 2002, Stat. Comput..