Tests for high-dimensional data based on means, spatial signs and spatial ranks
基于均值、空间符号与空间秩的高维数据检验AI 解读 ↓
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基于均值、空间符号与空间秩的高维数据检验
- 研究问题
- 当维度超过样本量时，均值检验与空间符号、空间秩检验的功效如何受坐标依赖和尾部厚度影响，检验尺寸是否可靠？
- 方法与思路
- 作者比较单样本零均值与双样本均值相等检验，在不同坐标依赖及维度、样本量增长方式下推导功效的渐近性质。
- 材料陈述的贡献
- 摘要称适当混合条件下各类检验极限功效相同且不受尾厚影响；更强依赖下符号和秩检验可更强，并报告尺寸比较。
- 阅读关注点
- 应核对备择序列、坐标混合条件及强依赖设定，区分仅维度增长与样本量同时增长，并检查渐近近似的尺寸表现。
- 阅读边界
- 仅据未注明稿件版本的OpenAlex索引摘要；尾部无关与功效优势均依赖所述渐近条件，不能推广为无条件稳健或占优。
原文摘要
Tests based on mean vectors and spatial signs and ranks for a zero mean in one-sample problems and for the equality of means in two-sample problems have been studied in the recent literature for high-dimensional data with the dimension larger than the sample size. For the above testing problems, we show that under suitable sequences of alternatives, the powers of the mean-based tests and the tests based on spatial signs and ranks tend to be same as the data dimension tends to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions. Further, their limiting powers do not depend on the heaviness of the tails of the distributions. This is in striking contrast to the asymptotic results obtained in the classical multivariate setting. On the other hand, we show that in the presence of stronger dependence among the coordinate variables, the spatial-sign- and rank-based tests for high-dimensional data can be asymptotically more powerful than the mean-based tests if, in addition to the data dimension, the sample size also tends to infinity. The sizes of some mean-based tests for high-dimensional data studied in the recent literature are observed to be significantly different from their nominal levels. This is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics. However, our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets.