In this work, we introduce a three-step semiparametric methodology for the estimation of production frontiers. We consider a model inspired by the well-known Cobb-Douglas production function, wherein input factors operate multiplicatively within the model. Efficiency in the proposed model is assumed to follow a continuous univariate uniparametric distribution in $(0,1)$, referred to as Matsuoka's distribution, which is discussed in detail. Following model linearization, the first step is to semiparametrically estimate the regression function through a local linear smoother. The second step focuses on the estimation of the efficiency parameter. Finally, we estimate the production frontier through a plug-in methodology. We present a rigorous asymptotic theory related to the proposed three-step estimation, including consistency, and asymptotic normality, and derive rates for the convergences presented. Incidentally, we also study the Matsuoka's distribution, deriving its main properties. The Matsuoka's distribution exhibits a versatile array of shapes capable of effectively encapsulating the typical behavior of efficiency within production frontier models. To complement the large sample results obtained, a Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed three-step methodology. An empirical application using a dataset of Danish milk producers is also presented.

翻译：本文提出了一种用于估计生产前沿的三步半参数方法。我们考虑一个受经典柯布-道格拉斯生产函数启发的模型，其中投入要素在模型中以乘法方式运作。假设所提出模型中的效率遵循定义在区间(0,1)上的连续单变量单参数分布，该分布被称为松冈分布，并对其进行了详细讨论。在模型线性化之后，第一步通过局部线性平滑器对回归函数进行半参数估计；第二步专注于效率参数的估计；最后通过插值法估计生产前沿。我们针对所提出的三步估计法建立了严谨的渐近理论，包括一致性与渐近正态性，并推导了所涉及收敛的速度。此外，我们还研究了松冈分布，推导了其主要性质。松冈分布展现出灵活多样的形状，能够有效刻画生产前沿模型中效率的典型行为。为了补充大样本结果，我们进行了蒙特卡洛模拟研究以评估所提三步法在有限样本下的表现，并利用丹麦牛奶生产商的数据集进行了实证应用。