Isotonic regression provides a flexible, tuning-free approach to estimating monotonic functions without imposing global curvature constraints, yet the estimated regression function is inherently a step function. This paper addresses a key limitation of such estimators: their inability to provide meaningful marginal properties, such as shadow prices or elasticities. We propose a novel piece-wise linear smoothing framework that recovers meaningful marginal estimates even in non-convex settings. Building on the concept of conditional convexity originally developed in deterministic frontier analysis, we formulate the smoothing process as a bilevel optimization problem that fits a continuous, monotonic, piece-wise linear function to the initial isotonic regression predictions. Monte Carlo simulations demonstrate that the proposed approach can significantly improve estimation precision, reducing mean squared error in both convex and non-convex settings for univariate and multivariate data. We apply this approach to analyze agglomeration economies in Finnish municipalities, illustrating its practical value.

翻译：保序回归提供了一种灵活、无需调参的方法来估计单调函数，且无需施加全局曲率约束，但估计得到的回归函数本质上是阶梯函数。本文解决了此类估计量的一大关键局限：无法提供有意义的边际性质（如影子价格或弹性）。我们提出了一种新颖的分段线性平滑框架，即使在非凸设定下也能恢复有意义的边际估计。基于最初在确定性前沿分析中发展的条件凸性概念，我们将平滑过程构建为双层优化问题，旨在拟合一条连续、单调、分段线性函数至初始保序回归预测结果。蒙特卡洛模拟表明，所提出的方法能显著提升估计精度，在单变量和多变量数据的凸与非凸设定下均能降低均方误差。我们将该方法应用于分析芬兰城市的集聚经济效应，展示了其实际应用价值。