Some Extensions of Tukeyʹs Depth Function

As the extensions of Tukeyʹs 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. Such kind ordering can retain more information from the original data than that based on Tukeyʹs 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 asymptotic breakdown points at least 1/3 and convergence rates of 1/n. Their asymptotic distributions are also obtained under some regularity conditions. A new algorithm based on the idea of thresholding is presented for computing these kinds of estimates and realized in the bivariate case. Simulations indicate that some of them could have the empirical mean squared errors smaller than those based on Tukeyʹs depth function or Donohoʹs depth function.