¹ Department of Economics, Yale University, New Haven, CT 06520, USA ² The Guanghua School of Management, Peking University, Beijing 100871, China ³ School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China ⁴ Department of Economics, New York University, New York, NY 10012, USA

This paper considers the estimation of an unknown function $h$ that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator.

The first anther was supported by US National Science Foundation (Grant No. SES-0631613) and the Cowles Foundation for Research in Economics