Influence functions for a general class of depth-based generalized quantile functions

Given a multivariate probability distribution F, a corresponding depth function orders points according to their “centrality” in the distribution F. One useful role of depth functions is to generate two-dimensional curves for convenient and practical description of particular features of a multivariate distribution, such as dispersion and kurtosis. Here the robustness of sample versions of such curves is explored via the influence function approach applied to the relevant functionals, using structural representations of the curves as generalized quantile functions. In particular, for a general class of so-called Type D depth functions including the well-known Tukey or halfspace depth, we obtain influence functions for the depth function itself, the depth distribution function, the depth quantile function, and corresponding depth-based generalized quantile functions. Robustness behavior similar to the usual univariate quantiles is found and quantified: the influence functions are of step function form with finite gross error sensitivity but infinite local shift sensitivity. Applications to a “scale” curve, a Lorenz curve for “tailweight”, and a “kurtosis” curve are treated. Graphical illustrations are provided for the influence functions of the scale and kurtosis curves in the case of the bivariate standard normal distribution and the halfspace depth function.