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Title: A note on the robustness of multivariate medians
Authors: Chakraborty, B. 
Chaudhuri, P.
Keywords: Affine equivariance
Asymptotic efficiency
Breakdown point
Multivariate location
Transformation retransformation estimate
Issue Date: 15-Nov-1999
Citation: Chakraborty, B.,Chaudhuri, P. (1999-11-15). A note on the robustness of multivariate medians. Statistics and Probability Letters 45 (3) : 269-276. ScholarBank@NUS Repository.
Abstract: In this note we investigate the extent to which some of the fundamental properties of univariate median are retained by different multivariate versions of median with special emphasis on robustness and breakdown properties. We show that transformation retransformation medians, which are affine equivariant, n1/2-consistent and asymptotically normally distributed under standard regularity conditions, can also be very robust with high breakdown points. We prove that with some appropriate adaptive choice of the transformation matrix based on a high breakdown estimate of the multivariate scatter matrix (e.g. S-estimate or minimum covariance determinant estimate), the finite sample breakdown point of a transformation retransformation median will be as high as n-1[(n-d+1)/2], where n= the sample size, d= the dimension of the data, and [x] denotes the largest integer smaller than or equal to x. This implies that as n→∞, the asymptotic breakdown point of a transformation retransformation median can be made equal to 50% in any dimension just like the univariate median. We present a brief comparative study of the robustness properties of different affine equivariant multivariate medians using an illustrative example. © 1999 Elsevier Science B.V.
Source Title: Statistics and Probability Letters
ISSN: 01677152
Appears in Collections:Staff Publications

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