Some linear programming methods for frontier estimation
Résumé
We propose new methods for estimating the frontier of a set of points. The estimates are defined as kernel functions covering all the points and whose associated support is of smallest surface. They are written as linear combinations of kernel functions applied to the points of the sample. The weights of the linear combination are then computed by solving a linear programming problem. In the general case, the solution of the optimization problem is sparse, that is, only a few coefficients are non-zero. The corresponding points play the role of support vectors in the statistical learning theory. In the case of uniform bivariate densities, the L1 error between the estimated and the true frontiers is shown to be almost surely converging to zero, and the rate of convergence is provided. The behaviour of the estimates on one finite sample situation is illustrated on simulations.