ABSTRACT Production frontier is an important concept in modern economics and has been widely used to measure production efficiency. Existing nonparametric frontier models often only allow one or low-dimensional input variables due to ‘curse-of-dimensionality’. In this paper we propose a flexible additive frontier model which quantifies the effects of multiple input variables on the maximum output. In addition, we consider the estimation of the nonparametric frontier functions with shape restrictions. Economic theory often imposes shape constraints on production frontier, such as, monotonicity and concavity. A two-step constrained polynomial spline method is proposed to give smooth estimates that automatically satisfy such shape constraints. The proposed method is not only easy to compute, but also more robust to outliers. In theory, we established uniform consistency of the proposed method. We illustrate the proposed method by both simulation studies and an application to the Norwegian farm data. The numerical studies suggest that the proposed method has superior performance by incorporating shape constraints.

中文翻译：

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具有形状约束的加性前沿函数的估计

摘要 生产前沿是现代经济学中的一个重要概念，被广泛用于衡量生产效率。由于“维度诅咒”，现有的非参数前沿模型通常只允许一维或低维输入变量。在本文中，我们提出了一个灵活的附加前沿模型，该模型量化了多个输入变量对最大输出的影响。此外，我们考虑了具有形状限制的非参数前沿函数的估计。经济理论经常对生产边界施加形状约束，例如单调性和凹面性。提出了一种两步约束多项式样条方法，以给出自动满足此类形状约束的平滑估计。所提出的方法不仅易于计算，而且对异常值更稳健。理论上，我们建立了所提出方法的统一一致性。我们通过模拟研究和对挪威农场数据的应用来说明所提出的方法。数值研究表明，所提出的方法通过结合形状约束具有优越的性能。