Least Squares estimation of two ordered monotone regression curves

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) = ∑n j=1(yj − aj)2w1 j + ∑n j=1(zj − bj)2w2 j over the class of pairs of vectors (a, b) ∈ R × R 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.