Central limit theorems for linear spectral statistics of large dimensional F-matrices

Shurong Zheng

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 444-476
  • ISSN: 0246-0203

Abstract

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In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the F matrix. In this paper, we establish the central limit theorems with explicit expressions of means and covariance functions for the linear spectral statistics of the large dimensional F matrix, where the dimension p of the two samples tends to infinity proportionally to the sample sizes (n1, n2). Moreover, the assumptions of the i.i.d. structures of arrays (Xjk)p×n1, (Yjk)p×n2 and the restriction of the fourth moments equaling 2 or 3 made in Bai and Silverstein (Ann. Probab.32 (2004) 553–605) are relaxed to that arrays (Xjk)p×n1 and (Yjk)p×n2 are independent respectively but not necessarily identically distributed except for a common fourth moment for each array. As a consequence, we obtain the central limit theorems for the linear spectral statistics of the beta matrix that is of the form (I + d ⋅ F matrix)−1, where d is a constant and I is an identity matrix.

How to cite

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Zheng, Shurong. "Central limit theorems for linear spectral statistics of large dimensional F-matrices." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 444-476. <http://eudml.org/doc/272099>.

@article{Zheng2012,
abstract = {In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the F matrix. In this paper, we establish the central limit theorems with explicit expressions of means and covariance functions for the linear spectral statistics of the large dimensional F matrix, where the dimension p of the two samples tends to infinity proportionally to the sample sizes (n1, n2). Moreover, the assumptions of the i.i.d. structures of arrays (Xjk)p×n1, (Yjk)p×n2 and the restriction of the fourth moments equaling 2 or 3 made in Bai and Silverstein (Ann. Probab.32 (2004) 553–605) are relaxed to that arrays (Xjk)p×n1 and (Yjk)p×n2 are independent respectively but not necessarily identically distributed except for a common fourth moment for each array. As a consequence, we obtain the central limit theorems for the linear spectral statistics of the beta matrix that is of the form (I + d ⋅ F matrix)−1, where d is a constant and I is an identity matrix.},
author = {Zheng, Shurong},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {linear spectral statistics; central limit theorem; F-matrix; beta matrix; Fisher matrix; covariance matrices},
language = {eng},
number = {2},
pages = {444-476},
publisher = {Gauthier-Villars},
title = {Central limit theorems for linear spectral statistics of large dimensional F-matrices},
url = {http://eudml.org/doc/272099},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Zheng, Shurong
TI - Central limit theorems for linear spectral statistics of large dimensional F-matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 444
EP - 476
AB - In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the F matrix. In this paper, we establish the central limit theorems with explicit expressions of means and covariance functions for the linear spectral statistics of the large dimensional F matrix, where the dimension p of the two samples tends to infinity proportionally to the sample sizes (n1, n2). Moreover, the assumptions of the i.i.d. structures of arrays (Xjk)p×n1, (Yjk)p×n2 and the restriction of the fourth moments equaling 2 or 3 made in Bai and Silverstein (Ann. Probab.32 (2004) 553–605) are relaxed to that arrays (Xjk)p×n1 and (Yjk)p×n2 are independent respectively but not necessarily identically distributed except for a common fourth moment for each array. As a consequence, we obtain the central limit theorems for the linear spectral statistics of the beta matrix that is of the form (I + d ⋅ F matrix)−1, where d is a constant and I is an identity matrix.
LA - eng
KW - linear spectral statistics; central limit theorem; F-matrix; beta matrix; Fisher matrix; covariance matrices
UR - http://eudml.org/doc/272099
ER -

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