Estimating a Function and Its Derivatives Under a Smoothness Condition
研究如何从含噪声数据中估计未知光滑函数及其偏导数，提出两种可计算的二次规划估计量，证明其一致性和收敛速度，并用股票期权定价例子展示效果。
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.