The work of A. Nazin has been carried out during his stay in INRIA Rhône-Alpes as invited professor, November–December 2002.
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Research Article
Some linear programming methods for frontier estimation
Article first published online: 23 MAR 2005
DOI: 10.1002/asmb.535
Copyright © 2005 John Wiley & Sons, Ltd.
Issue
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Applied Stochastic Models in Business and Industry
Special Issue: Statistical Learning
Volume 21, Issue 2, pages 175–185, March/April 2005
Additional Information
How to Cite
Bouchard, G., Girard, S., Iouditski, A. and Nazin, A. (2005), Some linear programming methods for frontier estimation. Appl. Stochastic Models Bus. Ind., 21: 175–185. doi: 10.1002/asmb.535
Publication History
- Issue published online: 23 MAR 2005
- Article first published online: 23 MAR 2005
Funded by
- Federal Office for Scientific, Technical and Cultural Affairs. Grant Number: P5/24
- Abstract
- References
- Cited By
Keywords:
- functional estimate;
- linear programming;
- L1 error;
- frontier estimation
Abstract
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. Copyright © 2005 John Wiley & Sons, Ltd.