Likelihood Ratio Tests for High‐Dimensional Normal Distributions
Likelihood Ratio Tests for High‐Dimensional Normal Distributions
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DOI：
10.1111/sjos.12147
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发表时间：
2013-06
影响因子：
1
通讯作者：
Tiefeng Jiang;F. Yang
中科院分区：
文献类型：
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作者：
Tiefeng Jiang;F. Yang
在最近的工作中，Jiang 和 Yang 研究了高维环境下的六种经典似然比检验统计量。假设从 p 维正态总体中观察到大小为 n 的随机样本，当 p 和 n 彼此成比例时，他们推导出中心极限定理（CLT），这与经典的卡方极限不同，因为 n 趋于无穷大，而 p 保持固定。在本文中，通过开发一种新工具，我们证明了上述六个 CLT 在更适用的设置中成立：p 趋于无穷大，并且 p 可以非常接近 n。这几乎是 CLT 的充分必要条件。随后介绍了直方图的模拟、大小和功效与经典卡方近似的比较以及讨论。
In their recent work, Jiang and Yang studied six classical Likelihood Ratio Test statistics under high‐dimensional setting. Assuming that a random sample of size n is observed from a p‐dimensional normal population, they derive the central limit theorems (CLTs) when p and n are proportional to each other, which are different from the classical chi‐square limits as n goes to infinity, while p remains fixed. In this paper, by developing a new tool, we prove that the mentioned six CLTs hold in a more applicable setting: p goes to infinity, and p can be very close to n. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi‐square approximations and discussions are presented afterwards.