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.

中文翻译：

---

与 ANOVA 分解模型相关的基于稳定雅可比多项式的最小二乘回归估计器

在这项工作中，我们构建了一个稳定且相当快速的估计器来解决非参数多维回归问题。所提出的估计器基于多元雅可比多项式的使用，该多项式为减小的$d-$variate 有限维多项式空间的大小提供了基础。ANOVA 分解技巧已用于构建这个后来的多项式空间。此外，通过使用正定随机矩阵理论的一些结果，我们证明了所提出的估计量在回归问题的不同协变量的独立同分布随机抽样点服从$d-维Beta分布的条件下是稳定的. 此外，我们为读者提供了估计器的 $L^2-$risk 误差的估计值。而且，在回归函数属于某个加权 Sobolev 空间的条件下，提供了对近似质量的更精确估计。最后，数值模拟支持了这项工作的各种理论结果。