Large deviation properties for empirical quantile-type production functions

Abstract

In standard microeconomic theory of the firm, the production frontier is used to describe the geometric locus of the optimal production. Most limitations of the conventional estimators of this efficient frontier stem from its reliance on estimation of the upper surface of the joint support of (X, Y), where X∈ℝ ^p ₊ represents the inputs-usage and Y∈ℝ₊ is the produced output. Instead, a recent approach involving the estimation of a partial quantile-type frontier of the joint support, lying close to its full boundary, has emerged in the literature as an attractive alternative. In this paper, we present a new and simple proof of the asymptotic normality of the resulting nonparametric frontier estimators and provide an estimate for the accuracy of the normal approximation. We also propose a Kiefer-type asymptotic representation for these estimators. The main results concern the study of some weak, moderate and strong large deviation properties of the quantile-based frontier estimators.

Keywords:

• quantile-based production frontiers
• probabilities of large deviations

2000 AMS Subject Classification :

• 62G20
• 60F10
• 91B38