Likelihood Ratio Tests for High-Dimensional Normal Distributions

2015 ◽  
Vol 42 (4) ◽  
pp. 988-1009 ◽  
Author(s):  
Tiefeng Jiang ◽  
Yongcheng Qi
Keyword(s):  
Likelihood Ratio ◽  
Likelihood Ratio Tests ◽  
High Dimensional ◽  
Normal Distributions

A study of two high-dimensional likelihood ratio tests under alternative hypotheses

2018 ◽  
Vol 07 (01) ◽  
pp. 1750016 ◽  
Author(s):  
Huijun Chen ◽  
Tiefeng Jiang
Keyword(s):  
Multivariate Analysis ◽  
Likelihood Ratio ◽  
Limit Theorems ◽  
Classical Problem ◽  
Likelihood Ratio Tests ◽  
High Dimensional ◽  
Ratio Test ◽  
Dimension Formula ◽  
Alternative Hypotheses ◽  
Distribution Testing

Let [Formula: see text] be a [Formula: see text]-dimensional normal distribution. Testing [Formula: see text] equal to a given matrix or [Formula: see text] equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension [Formula: see text] is fixed, it is known that the LRT statistics go to [Formula: see text]-distributions. When [Formula: see text] is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that [Formula: see text] and [Formula: see text] are proportional to each other. The condition [Formula: see text] suffices in our results.

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