We consider the problem of estimating an unknown function f* and its partial derivatives from a noisy data set of n observations, where we make no assumptions about f* 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 f* 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 n increases to infinity. 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.

翻译：我们考虑从包含n个观测值的含噪数据集中估计未知函数f*及其偏导数的问题，其中除假设f*满足具有m阶平方可积偏导数的光滑性条件外，不作任何其他假设。此类问题中f*的自然估计量是最符合数据集且满足特定光滑性条件的拟合函数。该估计量可视为在光滑性测度上界约束下的最小二乘估计量。另一种有效的估计量是在平均平方误差上界约束下最小化光滑性程度的估计量。我们证明这两种估计量均可表示为二次规划问题的解，建立了估计量及其偏导数的一致性，并研究了当n趋于无穷大时的收敛速度。数值实验验证了该估计方法在估计股票期权价值及其二阶导数（作为标的股票价格函数）时的有效性。