Spectral analysis of the Moore-Penrose inverse of a large dimensional sample covariance matrix

Taras Bodnar, Holger Dette, Nestor Parolya

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

Abstract

For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix σn let S~n denote the usual sample covariance (centered by the mean) and Sn the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p>n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore-Penrose inverse of Sn and S~n. We consider the large dimensional asymptotics when the number of variables p→∞ and the sample size n→∞ such that p/n→c∈(1, +∞). We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of S~n differs in the mean from that of Sn.

Original languageEnglish
Pages (from-to)160-172
Number of pages13
JournalJournal of Multivariate Analysis
Volume148
DOIs
Publication statusPublished - 1 Jun 2016
Externally publishedYes

Keywords

  • CLT
  • Large-dimensional asymptotics
  • Moore-Penrose inverse
  • Random matrix theory

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