Abstract

Some extensions of Tukey's depth function

J. ZHANG

As the extension of Tukey's depth, a family of affine invariant depth functions are introduced for multivariate location and dispersion. The location depth functions can be used for the purpose of multivariate ordering can retain more information from the original data than the based on Tukey's depth. The dispersion depth functions provide some additional view of the dispersion of the data set. It is shown that these sample depth functions converge to their population versions uniformly on any compact subset of the parameter space. The deepest points of these depth functions are affine equivariant estimates of multivariate location and dispersion. Under some general conditions these estimates are proved to have aymptotic breakdown points at least 1/3 and convergence rates of 1/. Their aymptotic distributions are also obtained under some regularity conditions. A new algorithm based on the idea of thresholding is presented for computing these kinds of them could have the empirical mean squared errors smaller than those based on Tukey's depth function of Donoho's depth function.

  Last update: October, 8, 2002  - Contact : S. Malali