Least Squares Estimation of Two Ordered Monotone Regression Curves

• Activity Number: 306
• Type: Contributed
• Date/Time: Tuesday, August 3, 2010 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Kaspar Rufibach*+ and Fadoua Balabdaoui and Filippo Santambrogio
• Companies: University of Zurich and Universite Paris-Dauphine and Universite Paris-Dauphine
• Keywords: least squares ; monotone regression ; pool-adjacent-violaters algorithm (PAVA) ; shape constraint estimation ; subgradient algorithm

Abstract

We consider the problem of finding the Least Squares estimators of two isotonic regression curves under the constraint that one lies above the other. The characterization of the estimators is established. Dykstra's algorithm is used to find the estimates for given data. In addition, a version of Shor's iterative projected subgradient algorithm is presented that is also applicable in more general settings than least squares. The projection in Shor's algorithm is computed via a generalized pool-adjacent-violaters algorithm, a byproduct of this work. Initially, the problem was motivated by stress-strain curve data from mechanical engineering. We illustrate our novel approach on this dataset. The proposed algorithms and the engineering dataset are available in the R-package OrdMonReg.