Functional convergence of quantile-type frontiers with application to parametric approximations

Nonparametric estimators of the upper boundary of the support of a multivariate distribution are very appealing because they rely on very few assumptions. But in productivity and efficiency analysis, this upper boundary is a production (or a cost) frontier and a parametric form for it allows for a richer economic interpretation of the production process under analysis. On the other hand, most of the parametric approaches rely on often too restrictive assumptions on the stochastic part of the model and are based on standard regression techniques fitting the shape of the center of the cloud of points rather than its boundary. To overcome these limitations, Florens and Simar [2005. Parametric approximations of nonparametric frontiers. J. Econometrics 124 (1), 91–116] propose a two-stage approach which tries to capture the shape of the cloud of points near its frontier by providing parametric approximations of a nonparametric frontier. In this paper we propose an alternative method using the nonparametric quantile-type frontiers introduced in Aragon, Daouia and Thomas-Agnan [2005. Nonparametric frontier estimation: a conditional quantile-based approach. Econometric Theory 21, 358–389] for the nonparametric part of our model. These quantile-type frontiers have the superiority of being more robust to extremes. Our main result concerns the functional convergence of the quantile-type frontier process. Then we provide convergence and asymptotic normality of the resulting estimators of the parametric approximation. The approach is illustrated through simulated and real data sets.