论文标题
限制了正常随机向量独立性的似然比测试统计数据的分布
Limiting distributions of the likelihood ratio test statistics for independence of normal random vectors
论文作者
论文摘要
考虑从$ p $变量的正常随机向量独立的可能性比率测试（LRT）统计数据。我们致力于基于尺寸$ n $的随机样本来得出LRT统计信息的限制分布。众所周知，当数据的尺寸或参数的数量固定时，极限是卡方分布。在Qi，Wang和Zhang（Ann Inst Stat Math（2019）71：911--946）的最新作品中，证明LRT统计数据在正常随机子矢量的长度相对平衡的条件下是渐近正常的，如果Dimension $ p $ to to dimension $ p $与样品大小$ n $ n $ n $ n $ n $ p $相对平衡。在本文中，我们研究了在一般条件下LRT统计量的限制分布。我们发现所有类型的限制分布，并为LRT统计量获得必要的和足够的条件，以在$ p $转移到无穷大时将其收敛为正态分布。我们还研究了Qi，Wang和Zhang（2019）中提出的调整后LRT测试统计量的限制分布。此外，我们提出了模拟结果，以比较经典卡方近似，正常和非正常近似与LRT统计的性能，卡方近似与调整后的测试统计量以及其他一些测试统计量。
Consider the likelihood ratio test (LRT) statistics for the independence of sub-vectors from a $p$-variate normal random vector. We are devoted to deriving the limiting distributions of the LRT statistics based on a random sample of size $n$. It is well known that the limit is chi-square distribution when the dimension of the data or the number of the parameters are fixed. In a recent work by Qi, Wang and Zhang (Ann Inst Stat Math (2019) 71: 911--946), it was shown that the LRT statistics are asymptotically normal under condition that the lengths of the normal random sub-vectors are relatively balanced if the dimension $p$ goes to infinity with the sample size $n$. In this paper, we investigate the limiting distributions of the LRT statistic under general conditions. We find out all types of limiting distributions and obtain the necessary and sufficient conditions for the LRT statistic to converge to a normal distribution when $p$ goes to infinity. We also investigate the limiting distribution of the adjusted LRT test statistic proposed in Qi, Wang and Zhang (2019). Moreover, we present simulation results to compare the performance of classical chi-square approximation, normal and non-normal approximation to the LRT statistics, chi-square approximation to the adjusted test statistic, and some other test statistics.
