高维多样本Behrens-Fisher问题的投影检验

High-Dimensional Multiple-Samples Projection Tests for the Behrens-Fisher Problem

摘要: 本文主要考虑多个高维总体均值向量是否相等的热点问题，大部分现有方法往往局限于同方差假设下，但在实际问题中该假设很难得到保证。利用Scheffe变换我们将多组样本构造成一组的新向量，并提出了一种投影检验方法以及它的渐进分布类似空间符号检验。该方法利用权函数和样本分割方法减少功效损失和保持独立性，并基于最优势得出最优投影方向和权函数，且可以推广到部分非正态分布。最后，我们进行蒙特卡洛实验验证它在有限样本下的检验效率，结果表明新检验在高相关性和一定非正态性假设下优于已有测试。

Abstract: This paper mainly considers the hot topic of whether multiple high-dimensional population means are equal. Most of the existing methods are often limited to the homoskedasticity hypothesis, which is difficult to be guaranteed in practice. By Scheffe’s transformation, we transform multiple samples to a group of new vectors, and propose a projection test and its asymptotic distribution similar to the spatial sign test. This method uses weight function and sample-splitting strategy to reduce power loss and maintain independence, and give the optimal projection direction and weight func-tion based on the optimal power, furthermore it can apply to some non-normal distributions. Finally, we conduct Monte Carlo simulations to examine the finite sample performance, and the results show that the new test outperforms the existing test under high correlation and some non-normal assumptions.