Spiked eigenvalues of noncentral Fisher matrix with applications

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2021-04-10 DOI:10.3150/22-bej1579
Xiaozhuo Zhang, Zhiqiang Hou, Z. Bai, Jiang Hu
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引用次数: 4

Abstract

In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$, where ${\mathbf C}_n$ is a noncentral sample covariance matrix defined by $(\mathbf \Xi+\mathbf X)(\mathbf \Xi+\mathbf X)^*/n$ and $\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$. The matrices $\mathbf X$ and $\mathbf Y$ are two independent {Gaussian} arrays, with respective $p\times n$ and $p\times N$ and the Gaussian entries of them are \textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $n$, and $N$ grow to infinity proportionally, we establish a phase transition of the spiked eigenvalues of $\mathbf F_p$. Furthermore, we derive the \textit{central limiting theorem} (CLT) for the spiked eigenvalues of $\mathbf F_p$. As an accessory to the proof of the above results, the fluctuations of the spiked eigenvalues of ${\mathbf C}_n$ are studied, which should have its own interests. Besides, we develop the limits and CLT for the sample canonical correlation coefficients by the results of the spiked noncentral Fisher matrix and give three consistent estimators, including the population spiked eigenvalues and the population canonical correlation coefficients.
非中心Fisher矩阵的尖峰特征值及其应用
本文研究了由${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$定义的非中心Fisher矩阵的峰值特征值的渐近性,其中${\mathbf C}_n$是由$(\mathbf \Xi+\mathbf X)(\mathbf \Xi+\mathbf X)^*/n$和$\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$定义的非中心样本协方差矩阵。矩阵$\mathbf X$和$\mathbf Y$是两个独立的{高斯}数组,分别为$p\times n$和$p\times N$,它们的高斯项为\textit独立同分布{(i.i.d),均值}$0$,方差$1$。当$p$, $n$和$N$按比例增长到无穷大时,我们建立了$\mathbf F_p$的尖峰特征值的相变。进一步,我们导出了$\mathbf F_p$的尖征值的\textit{中心极限定理}(CLT)。作为证明上述结果的辅助,研究了${\mathbf C}_n$的尖刺特征值的波动,这应该有它自己的兴趣。此外,利用尖刺非中心Fisher矩阵的结果,给出了样本典型相关系数的极限和CLT,并给出了种群尖刺特征值和种群典型相关系数的三个一致估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
期刊介绍: BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
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