A three-step approach to production frontier estimation and the Matsuoka's distribution

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.