Estimating a Function and Its Derivatives Under a Smoothness Condition

We consider the problem of estimating an unknown function [Formula: see text] and its partial derivatives from a noisy data set of n observations, where we make no assumptions about [Formula: see text] except that it is smooth in the sense that it has square integrable partial derivatives of order m. A natural candidate for the estimator of [Formula: see text] in such a case is the best fit to the data set that satisfies a certain smoothness condition. This estimator can be seen as a least squares estimator subject to an upper bound on some measure of smoothness. Another useful estimator is the one that minimizes the degree of smoothness subject to an upper bound on the average of squared errors. We prove that these two estimators are computable as solutions to quadratic programs, establish the consistency of these estimators and their partial derivatives, and study the convergence rate as [Formula: see text]. The effectiveness of the estimators is illustrated numerically in a setting where the value of a stock option and its second derivative are estimated as functions of the underlying stock price.