A note on quantiles for multidimensional data

In many economic applications, we need to compute multidimensional quantiles or to estimate the center of multidimensional distribution. Our approach to multidimensional quantiles is based on a modified version of interdirections which are given by halving hyperplanes. The methods working with halving hyperplanes do not require any knowledge about the location of the distribution, what is their main advantage in comparison with classical approaches. Our aim is to study asymptotics for multidimensional quantiles. Under our approach to multidimensional quantiles, the very important tool for studying of these asymptotics are U-statistics. In fact, we need to study asymptotics for U-statistics (especially, weighted U-statistics and incomplete U-statistics).