Least Squares estimation of two ordered monotone regression curves

In this paper, we consider the problem of estimating two monotone regression curves $g^\circ_1$ and $g^\circ_2$ under the additional constraint that they are ordered; e.g., $g^\circ_1 \ge g^\circ_2$. Here, we assume that the true regression curves are antitonic. Given two sets of $n$ data points $y_1, .., y_n$ and $z_1, ...,z_n$ that are observed at (the same) deterministic points $x_1, ..., x_n$, the estimates are obtained by minimizing the Least Squares criterion $L_2(f_1, f_2) = \sum_{j=1}^n (y_j - f_1(x_j))^2 w_1(x_j) + sum_{j=1}^n (z_j - f_2(x_j))^2 w_2(x_j)$ over the class of pairs of functions $(f_1, f_2)$ such that $f_1$ and $f_2$ are antitonic and $f_1(x_j) \ge f_2(x_j)$ for all $j \in \{1, ..., n\}$. 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.