INTRODUCTION

The development of nonparametric estimation in econometrics has been extremely important over the past fifteen years. Inference was first concentrated on the data's distribution, described, for example, by its density or by its hazard function, or by some characteristics of the conditional distributions, such as the conditional expectations. This approach is typically a reduced-form analysis oriented to sophisticated data description, even if the selection of conditioning variables may depend on a theoretical model. On the other side, the structural econometric analysis is focused on the estimation of the (possibly functional) parameters that describe the economic agent's behavior and that are not, in general, “simple” transformations of the sampling distribution. An excellent state-of-the-art discussion of nonparametric econometrics is given by Pagan and Ullah (1999) (see also Aït Sahalia (1995) for a general “reduced form analysis” in the nonparametric framework).

A first objective of this paper is to introduce a general framework for structural functional inference in connection with the inverse-problems literature. An inverse problem is the resolution of a functional equation with a particular attention to the sensitivity of the solution to possible errors in the specification of the equation due for instance to an estimation procedure (see, e.g., for recent surveys of the literature, Colton et al., 2000).

More specifically, we analyze linear inverse problems in which the parameter of interest is a function ϕ solution of a linear equation KFϕ = ψF in which both the linear operator KF and the right-hand side depend on the (unknown) distribution F of the sample.