## Abstract

We establish the validity of an Edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The result covers ARFIMA-type models, including fractional Gaussian noise. The method of proof consists of three main ingredients: (i) verification of a suitably modified version of Durbin's general conditions for the validity of the Edgeworth expansion to the joint density of the log-likelihood derivatives; (ii) appeal to a simple result of Skovgaard to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of Bhattacharya and Ghosh to accomplish the passage from the result on the log-likelihood derivatives to the result for the maximum likelihood estimators. We develop and make extensive use of a uniform version of a theorem of Dahlhaus on products of Toeplitz matrices, the extension of Dahlhaus' result is of interest in its own right. A small numerical study of the efficacy of the Edgeworth expansion is presented for the case of fractional Gaussian noise.

Original language | American English |
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Pages (from-to) | 586-612 |

Number of pages | 27 |

Journal | Annals of Statistics |

Volume | 31 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2003 |

## Keywords

- ARFIMA models
- Edgeworth expansions
- Long memory processes