A large amount of literature has been developed on how to specify and to estimate production frontiers or cost functions. Two different approaches have been mainly developed: the deterministic frontier model which relies on the assumption that all the observations are on a unique side of the frontier, and the stochastic frontier models where observational errors or random noise allows some observations to be outside of the frontier. In a deterministic frontier framework, nonparametric methods are based on envelopment techniques known as FDH (Free Disposal Hull) and DEA (Data Envelopment Analysis). Today, statistical inference based on DEA/FDH type of estimators is available but, by construction, they are very sensitive to extreme values and to outliers. In this paper, we build an original nonparametric estimator of the "efficient frontier" which is more robust to extreme values, noise or outliers than the standard DEA/FDH nonparametric estimators. It is based on a concept of expected minimum input function (or expected maximal output function). We show how these functions are related to the efficient frontier itself. The resulting estimator is also related to the FDH estimator but our estimator will not envelop all the data. The asymptotic theory is also provided. Our approach includes the multiple inputs and multiple outputs cases. As an illustration, the methodology is applied to estimate the expected minimum cost function for french post offices.

and Toulouse. We also thank R. Lestringand who performed the numerical illustration given in Section 5. 2 Lyxor Asset Management and CREST.

In other words, while the solvability criterion of Kϕ = r imposes r ∈ Ψ for some γ ≥ 1, we will get a geometric rate of decay of the regularization bias as soon as r ∈ Ψ for some γ > 1. Note that the upper bound β = 2 of the rate of decay is just a matter of normalization of the parameter α of regularization. By (3.19), ϕ − ϕ © cannot go to zero faster than [1 − q (α, λ)] ©

This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.