论文标题
稳定的Jacobi多项式基于最小二乘回归估计器与ANOVA分解模型相关
A Stable Jacobi polynomials based least squares regression estimator associated with an ANOVA decomposition model
论文作者
论文摘要
在这项工作中，我们构建了一个稳定且相当快的估计器，用于解决非参数多维回归问题。所提出的估计器基于使用多元雅各比多项式的使用，该多项式多项式为减小$ d- $差异有限尺寸多项式空间的基础。 ANOVA分解技巧已用于构建以后的多项式空间。同样，通过利用正定随机矩阵理论的一些结果，我们表明所提出的估计量在I.I.D.的条件下是稳定的。回归问题的不同协变量的随机采样点，遵循$ d- $尺寸beta分布。此外，我们还为读者提供了估计器的$ l^2- $风险错误的估计。此外，在回归函数属于某些加权Sobolev空间的条件下，提供了对近似质量的更精确估计。最后，这项工作的各种理论结果得到了数值模拟的支持。
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.
