In this work, we construct a stable and fairly fast estimator for solving non-parametric multidimensional regression problems. The proposed estimator is based on the use of multivariate Jacobi polynomials that generate a basis for a reduced size of $d-$variate finite dimensional polynomial space. An ANOVA decomposition trick has been used for building this later polynomial space. Also, by using some results from the theory of positive definite random matrices, we show that the proposed estimator is stable under the condition that the i.i.d. random sampling points for the different covariates of the regression problem, follow a $d-$dimensional Beta distribution. Also, we provide the reader with an estimate for the $L^2-$risk error of the estimator. Moreover, a more precise estimate of the quality of the approximation is provided under the condition that the regression function belongs to some weighted Sobolev space. Finally, the various theoretical results of this work are supported by numerical simulations.
翻译:在这项工作中,我们构建了一个稳定且相当快速的估算器,用于解决非参数性多维回归问题。 拟议的估算器基于使用多变量的 Jacobi 多元分子, 从而产生一个基础, 用于缩小 $d- $ variate 有限维度多面空间的大小。 一个 ANOVA 分解技巧已被用于构建这个后来的多元空间。 此外, 通过使用肯定的随机矩阵理论的某些结果, 我们显示, 拟议的估算器在以下条件下是稳定的: 回归问题不同共变量的 i. d. 随机取样点, 遵循 $d- d 的 Beta 分布 。 此外, 我们向读者提供估算器的 $L 2- $ 风险错误的估计数。 此外, 在回归功能属于某些加权的 Sobolev 空间的条件下, 提供了对近似质量的更精确的估计 。 最后, 这项工作的各种理论结果得到数字模拟的支持 。