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We consider the estimation of the boundary of a set when it is known to be sufficiently smooth, to satisfy certain shape constraints and to have an additive structure. Our proposed method is based on spline estimation of a conditional quantile regression and is resistant to outliers and/or extreme values in the data. This work is a desirable extension of existing works in the literature and can also be viewed as an alternative to existing estimators that have been used in empirical analysis. The results of a Monte Carlo study show that the new method outperforms the existing methods when outliers or heterogeneity are present. Our theoretical analysis indicates that our proposed boundary estimator is uniformly consistent under a set of standard assumptions. We illustrate practical use of our method by estimating two production functions using real-world datasets.

Robust Estimation of Additive Boundaries With Quantile Regression and Shape Constraints