Least Squares estimation of two ordered monotone regression curves

Abstract

In this paper, we consider the problem of finding the Least Squares estimators of two isotonic regression curves $g_1^{\circ }$ and $g_2^{\circ }$ under the additional constraint that they are ordered; e.g., $g_1^{\circ }\le g_2^{\circ }$. Given two sets of $n$ data points $y_1,...,y_n$ and $z_1,>...,z_n$ observed at (the same) design points, the estimates of the true curves are obtained by minimizing the weighted Least Squares criterion $L_2(a,b)={\sum }_{j=1}^n(y_j-a_j{)}^2{w}_{1,j}+{\sum }_{j=1}^n(z_j-b_j{)}^2{w}_{2,j}$ over the class of pairs of vectors $(a,b)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ such that $a_1\le a_2\le ...\le a_n$, $b_1\le b_2\le ...\le b_n$, and $a_i\le b_i,i=1,...,n$. 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.

23 pages, 2 figures. Second revised version according to reviewer comments

References in corpus (4)

• Inconsistency of bootstrap: The Grenander estimator
• Bootstrapping the Grenander estimator
• Maximum likelihood estimation of a multidimensional log-concave density
• Least Squares and Shrinkage Estimation under Bimonotonicity Constraints

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