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 introduced and explored. Following model linearization, the first step of the procedure is to semiparametrically estimate the regression function through a local linear smoother. The second step focuses on the estimation of the efficiency parameter in which the properties of the Matsuoka's distribution are employed. 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 introduce and study the Matsuoka's distribution, deriving its main properties, including quantiles, moments, $\alpha$-expectiles, entropies, and stress-strength reliability, among others. 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：本文提出了一种三步半参数方法用于生产前沿估计。我们考虑一个基于经典柯布-道格拉斯生产函数的模型，其中投入要素在模型中呈乘性作用。假设所提模型中的效率服从$(0,1)$区间上的连续单变量单参数分布，即松冈分布，本文对该分布进行了介绍与探讨。在模型线性化后，第一步通过局部线性平滑器对回归函数进行半参数估计；第二步利用松冈分布的性质来估计效率参数；最后，采用插入法估计生产前沿。我们建立了所提三步估计的严格渐近理论，包括相合性与渐近正态性，并推导了所呈现收敛性的收敛速率。同时，我们引入并研究了松冈分布，推导了其主要性质，包括分位数、矩、$\alpha$-期望分位数、熵以及应力-强度可靠性等。松冈分布具有多样化的形状，能够有效刻画生产前沿模型中效率的典型行为。为补充大样本结果，我们开展了蒙特卡洛仿真研究，以评估所提三步方法在有限样本下的表现。最后，本文还利用丹麦奶农数据集进行了实证应用。