# Applications of Peter Hall's martingale limit theory to estimating and testing high dimensional covariance matrices

Danning Li, Lingzhou Xue, Hui Zou

Research output: Contribution to journalArticle

1 Citation (Scopus)

### Abstract

Martingale limit theory is increasingly important in modern probability theory and mathematical statistics. In this article, we give a selected overview of Peter Hall's contributions to both the theoretical foundations and the wide applicability of martingales. We highlight his celebrated coauthored book, Hall and Heyde (1980) and his ground-breaking paper, Hall (1984). To illustrate the power of his martingale limit theory, we present two contemporary applications to estimating and testing high dimensional covariance matrices. In the first, we use the martingale central limit theorem in Hall and Heyde (1980) to obtain the simultaneous risk optimality and consistency of Stein's unbiased risk estimation (SURE) information criterion for large covariance matrix estimation. In the second application, we use the central limit theorem for degenerate U-statistics in Hall (1984) to establish the consistent asymptotic size and power against more general alternatives when testing high-dimensional covariance matrices.

Original language English (US) 2657-2670 14 Statistica Sinica 28 4 https://doi.org/10.5705/ss.202017.0060 Published - Oct 2018

### Fingerprint

Martingale
Covariance matrix
High-dimensional
Testing
Martingale Central Limit Theorem
Covariance Matrix Estimation
Information Criterion
U-statistics
Probability Theory
Central limit theorem
Optimality
Statistics
Alternatives

### All Science Journal Classification (ASJC) codes

• Statistics and Probability
• Statistics, Probability and Uncertainty

### Cite this

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title = "Applications of Peter Hall's martingale limit theory to estimating and testing high dimensional covariance matrices",
abstract = "Martingale limit theory is increasingly important in modern probability theory and mathematical statistics. In this article, we give a selected overview of Peter Hall's contributions to both the theoretical foundations and the wide applicability of martingales. We highlight his celebrated coauthored book, Hall and Heyde (1980) and his ground-breaking paper, Hall (1984). To illustrate the power of his martingale limit theory, we present two contemporary applications to estimating and testing high dimensional covariance matrices. In the first, we use the martingale central limit theorem in Hall and Heyde (1980) to obtain the simultaneous risk optimality and consistency of Stein's unbiased risk estimation (SURE) information criterion for large covariance matrix estimation. In the second application, we use the central limit theorem for degenerate U-statistics in Hall (1984) to establish the consistent asymptotic size and power against more general alternatives when testing high-dimensional covariance matrices.",
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In: Statistica Sinica, Vol. 28, No. 4, 10.2018, p. 2657-2670.

Research output: Contribution to journalArticle

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AU - Li, Danning

AU - Xue, Lingzhou

AU - Zou, Hui

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