### Abstract

Suppose that X_{n}=(X_{1},... X_{n}) is a collection of m-dimensional random vectors X_{i} forming a stochastic process with a parameter θ{symbol}. Let {Mathematical expression} be the MLE of θ{symbol}. We assume that a transformation A( {Mathematical expression}) of {Mathematical expression} has the k-thorder Edgeworth expansion (k=2,3). If A extinguishes the terms in the Edgeworth expansion up to k-th-order (k≥2), then we say that A is the k-th-order normalizing transformation. In this paper, we elucidate the k-th-order asymptotics of the normalizing transformations. Some conditions for A to be the k-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study the k-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.

Original language | English |
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Pages (from-to) | 581-600 |

Number of pages | 20 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 47 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1995 Sep 1 |

### Keywords

- Edgeworth expansion
- MLE
- Normalizing transformation
- higher-order asymptotic theory
- multivariate analysis
- observed information
- saddlepoint expansion
- signed log likelihood ratio
- time series analysis
- variance stabilizing transformation

### ASJC Scopus subject areas

- Statistics and Probability