Asymptotic locations of bounded and unbounded eigenvalues of sample correlation matrices of certain factor models -- application to a components retention rule

2024/07/10 by Yohji Akama, Akama, Yohji, Peng Tian +1
Computer Science · #FOS: Mathematics #Matrix Theory and Algorithms #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2407.07282

openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let the dimension N of data and the sample size T tend to ∞ with N/T → c > 0. The spectral properties of a sample correlation matrix C and a sample covariance matrix S are asymptotically equal whenever the population correlation matrix R is bounded (El Karoui 2009). We demonstrate this also for general linear models for unbounded R, by examining the behavior of the singular values of multiplicatively perturbed matrices. By this, we establish: Given a factor model of an idiosyncratic noise variance σ² and a rank-r factor loading matrix L which rows all have common Euclidean norm L. Then, the kth largest eigenvalues λ_k (1≤ k≤ N) of C satisfy almost surely: (1) λ_r diverges, (2) λ_k/s_k²→1/(L² + σ²) (1 ≤ k ≤ r) for the kth largest singular value s_k of L, and (3) λ_{r + 1}→(1-ρ)(1+√(c))² for ρ:= L²/(L² + σ²). Whenever s_r is much larger than √(log N), then broken-stick rule (Frontier 1976, Jackson 1993), which estimates rank L by a random partition (Holst 1980) of [0, 1], tends to r (a.s.). We also provide a natural factor model where the rule tends to "essential rank" of L (a.s.) which is smaller than rank L.

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