Papers › Least Squares estimation of two ordered monotone regression curves
Least Squares estimation of two ordered monotone regression curves
Fadoua Balabdaoui, Kaspar Rufibach, Filippo Santambrogio
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In this paper, we consider the problem of finding the Least Squares estimators of two isotonic regression curves g°₁ and g°₂ under the additional constraint that they are ordered; e.g., g°₁ ≤g°₂. Given two sets of n data points y₁, ..., yₙ and z₁, >...,zₙ observed at (the same) design points, the estimates of the true curves are obtained by minimizing the weighted Least Squares criterion L₂(a, b) = ∑ⱼ₌₁ⁿ (yⱼ - aⱼ)² w_(1,j)+ ∑ⱼ₌₁ⁿ (zⱼ - bⱼ)² w_(2,j) over the class of pairs of vectors (a, b) ∈ℝⁿ ×ℝⁿ such that a₁ ≤a₂ ≤...≤aₙ, b₁ ≤b₂ ≤...≤bₙ, and aᵢ ≤bᵢ, i=1, ...,n. The characterization of the estimators is established. To compute these estimators, we use an iterative projected subgradient algorithm, where the projection is performed with a "generalized" pool-adjacent-violaters algorithm (PAVA), a byproduct of this work. Then, we apply the estimation method to real data from mechanical engineering.
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