DETERMINANT OF SAMPLE CORRELATION MATRIX WITH APPLICATION

• 成果类型: Article
• 署名作者: Jiang, Tiefeng
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1362
• 发表日期: 2019
• 页码: 1356-1397
• 关键词: likelihood ratio tests; largest entries; LIMIT-THEOREMS; distributions

摘要

Let x(1 ), . . . , x(n) be independent random vectors of a common p-dimensional normal distribution with population correlation matrix R-n . The sample correlation matrix (R) over cap (n) = ((r) over cap (ij))(pxp) is generated from x(1) , . . . . ,x(n) such that (r) over cap (ij) is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x(1 ), . . . , x(n))'. The matrix (R) over cap (n) is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of (R) over cap (n) for a big class of R-n. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R-n is larger than 1/2. Besides, a formula of the moments of vertical bar(R) over cap (n)vertical bar and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.