Large sample approximation of the distribution for smooth monotone frontier estimation

Abdelaati Daouia (1) , Laurent Gardes (2) , Stéphane Girard (2)

1 GREMAQ - Groupe de recherche en économie mathématique et quantitative
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems

This paper deals with nonparametric estimation of the upper boundary of a multivariate support under monotonicity constraint. This estimation problem arises in various contexts such as efficiency and frontier analysis in econometrics and portfolio management. The traditional estimators based on envelopment techniques are very non-robust. To reduce this defect, previous works have rather concentrated on estimation of a concept of a partial frontier of order alpha ∈ (0, 1) lying near the full support boundary. However the resulting sample estimator is a discontinuous curve and suffers from a lack of efficiency due to the large variation of the extreme observations involved in its construction. A smoothed-kernel variant of this empirical estimator may be then preferable as shown recently in the econometric literature, but no attention was devoted to the limit distribution of the smoothed alpha-frontier when it estimates the true full boundary itself. In this paper, we address this problem by specifying the different limit laws of this estimator for fixed orders alpha ∈ (0, 1] as well as for sequences alpha depending on n tending to one at different rates as the sample size n goes to infinity.

Domains: Statistics Theory [stat.TH]; Statistics [math.ST]

Abdelaati Daouia, Laurent Gardes, Stéphane Girard. Large sample approximation of the distribution for smooth monotone frontier estimation. 2009. ⟨hal-00409447⟩