拟蒙特卡罗方法用于估计多元回归函数(英文)

Quasi-Monte Carlo Methods for Estimating a Multivariate Regression Function

拟蒙特卡罗（QMC）方法被广泛用于解决数值分析和统计学中的各种问题，比如数值积分，最优化，试验设计，随机过程的模拟等．本文研究该方法在估计多元回归函数中的应用．证明了，在相当一般的条件下，均匀设计（或者，“代表点设计”）与回归函数傅里叶系数的QMC估计（对应地，使用拟随机重要性抽样的QMC估计）一起，构成一个回归函数的渐近最优投影估计．

Quasi-Monte Carlo (QMC) methods are widely used for solving different problems of numerical analysis and statistics, such as integration, optimization, experimental design, simulation of stochastic processes, etc. In this paper, these methods are suggested for estimating a multivariate regression function. It is shown that, in a rather general case, the “uniform design”(respectively, the “representative points design”), together with standard QMC estimates (respetively,modified QMC estimates (using quasi-random importance sampling)) of Fourier coefficients of a regression function provide an asyinptotically optimal procedure of projection estimation of the regression function.