### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)

### Cite this

**Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators.** / Taniguchi, Masanobu; Puri, Madan L.

Research output: Contribution to journal › Article

*Annals of the Institute of Statistical Mathematics*, vol. 47, no. 3, pp. 581-600. https://doi.org/10.1007/BF00773402

}

TY - JOUR

T1 - Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators

AU - Taniguchi, Masanobu

AU - Puri, Madan L.

PY - 1995/9

Y1 - 1995/9

N2 - Suppose that Xn=(X1,... Xn) is a collection of m-dimensional random vectors Xi 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.

AB - Suppose that Xn=(X1,... Xn) is a collection of m-dimensional random vectors Xi 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.

KW - Edgeworth expansion

KW - higher-order asymptotic theory

KW - MLE

KW - multivariate analysis

KW - Normalizing transformation

KW - observed information

KW - saddlepoint expansion

KW - signed log likelihood ratio

KW - time series analysis

KW - variance stabilizing transformation

UR - http://www.scopus.com/inward/record.url?scp=0042105991&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042105991&partnerID=8YFLogxK

U2 - 10.1007/BF00773402

DO - 10.1007/BF00773402

M3 - Article

AN - SCOPUS:0042105991

VL - 47

SP - 581

EP - 600

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 3

ER -