# Ordered monotone regression

### Abstract

We consider the task of finding the Least Squares estimators of two isotonic regression curves $g_1$ and $g_2$ under the additional constraint that they are ordered, i.e, $g_1 <= g_2$. The characterization of the new estimators is established and algorithms to find them are discussed. Unfortunately, as in the one-curve case, the estimators are step-functions, i.e. non-continuous. To circumvent this problem we propose a kernel-smoothed version of the estimators. The chosen family of kernels entails that the smoothed estimates remain monotone. The problem was motivated by stress-strain curve data from a mechanical engineering. We illustrate the method on this dataset.

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Basel, Switzerland