### 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) |
---|---|

Pages (from-to) | 2657-2670 |

Number of pages | 14 |

Journal | Statistica Sinica |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1 2018 |

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### Keywords

- Degenerate U-statistics
- Hypothesis testing
- Large covariance matrix
- Martingale limit theory
- Stein's unbiased risk estimation

### Cite this

*Statistica Sinica*,

*28*(4), 2657-2670. https://doi.org/10.5705/ss.202017.0060

**Applications of Peter Hall's martingale limit theory to estimating and testing high dimensional covariance matrices.** / Li, Danning; Xue, Lingzhou; Zou, Hui.

Research output: Contribution to journal › Article

*Statistica Sinica*, vol. 28, no. 4, pp. 2657-2670. https://doi.org/10.5705/ss.202017.0060

}

TY - JOUR

T1 - Applications of Peter Hall's martingale limit theory to estimating and testing high dimensional covariance matrices

AU - Li, Danning

AU - Xue, Lingzhou

AU - Zou, Hui

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Degenerate U-statistics

KW - Hypothesis testing

KW - Large covariance matrix

KW - Martingale limit theory

KW - Stein's unbiased risk estimation

UR - http://www.scopus.com/inward/record.url?scp=85054552027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054552027&partnerID=8YFLogxK

U2 - 10.5705/ss.202017.0060

DO - 10.5705/ss.202017.0060

M3 - Article

AN - SCOPUS:85054552027

VL - 28

SP - 2657

EP - 2670

JO - Statistica Sinica

JF - Statistica Sinica

SN - 1017-0405

IS - 4

ER -