Statistica Sinica 8(1998), 827-839

CONSTRUCTING THE BIVARIATE TUKEY MEDIAN

IN ORDER-STATISTICS MODELS

Peter J. Rousseeuw and Ida Ruts

University of Antwerp

Abstract: The halfplane location depth of a point relative to a bivariate data set X={x₁,...,x_n} is the minimal number of observations in any closed halfplane that contains θ (Tukey (1975)). The halfplane median or Tukey median is the θ with maximal depth k (Donoho and Gasko (1992)). If this θ is not unique, the Tukey median is defined as the center of gravity of the set of points with depth k. In this paper we construct two algorithms for computing the Tukey median. The first one is relatively straightforward but quite slow, whereas the second (called HALFMED) is much faster. A small simulation study is performed, and some examples are given.

Key words and phrases: Algorithm, halfplane depth, robustness, Tukey depth.