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Linear programming problems for frontier estimation

Guillaume Bouchard 1 Stéphane Girard 1 Anatoli Juditsky 1 Alexander Nazin 2
1 IS2 - Statistical Inference for Industry and Health
Inria Grenoble - Rhône-Alpes, LBBE - Laboratoire de Biométrie et Biologie Evolutive - UMR 5558
Abstract : We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving a linear programming problem. In the general case, the solution of the optimizat- ion 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. The L_1 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 finite sample situations is illustrated on some simulations.
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Submitted on : Tuesday, May 23, 2006 - 7:03:40 PM
Last modification on : Friday, February 4, 2022 - 3:30:43 AM
Long-term archiving on: : Sunday, April 4, 2010 - 8:23:12 PM

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Guillaume Bouchard, Stéphane Girard, Anatoli Juditsky, Alexander Nazin. Linear programming problems for frontier estimation. [Research Report] RR-4717, INRIA. 2003. ⟨inria-00071869⟩

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