Asymptotically optimal representative points of bivariate random vectors

Optimal representative points (also called principal points) of scalar random variables are known in several cases and asymptotically optimal representative points are known for all densities, as the number of the points increases to infinity. When random quantities are uniformly distributed over a bounded two dimensional region, the centers of regular hexagons as representative points are asymptotically optimal. So far asymptotically optimal representative points of non-uniform multivariate distributions are not reported. Here, we give a method of designing representative points for non-uniform bivariate random vectors and show that the proposed method is asymptotically optimal. Examples of simulations with Gaussian, Pearson Type VII and Laplacian density functions are considered.