### Abstract

Some statistics in common use take the form of a ratio of two statistics, such as sample correlation coefficient, Pearson's coefficient of variation, cumulant estimators, and so on. In this article, using an asymptotic representation of the ratio statistics, we will obtain an Edgeworth expansion and a normalizing transformation with remainder term o(n-^{1/2}). The Edgeworth expansion is based on a Studentized ratio statistic, which is studentized by a consistent variance estimator. Applying these results to the sample correlation coefficient, we obtain the normalizing transformation and an asymptotic confidence interval of the correlation coefficient without assuming specific underlying distribution. This normalizing transformation is an extension of the Fisher's z-transformation.

Original language | English |
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Pages (from-to) | 1344-1358 |

Number of pages | 15 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 39 |

Issue number | 8-9 |

DOIs | |

Publication status | Published - Jan 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

**Edgeworth expansion and normalizing transformation of ratio statistics and their application.** / Maesono, Yoshihiko.

Research output: Contribution to journal › Article

*Communications in Statistics - Theory and Methods*, vol. 39, no. 8-9, pp. 1344-1358. https://doi.org/10.1080/03610920802311741

}

TY - JOUR

T1 - Edgeworth expansion and normalizing transformation of ratio statistics and their application

AU - Maesono, Yoshihiko

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Some statistics in common use take the form of a ratio of two statistics, such as sample correlation coefficient, Pearson's coefficient of variation, cumulant estimators, and so on. In this article, using an asymptotic representation of the ratio statistics, we will obtain an Edgeworth expansion and a normalizing transformation with remainder term o(n-1/2). The Edgeworth expansion is based on a Studentized ratio statistic, which is studentized by a consistent variance estimator. Applying these results to the sample correlation coefficient, we obtain the normalizing transformation and an asymptotic confidence interval of the correlation coefficient without assuming specific underlying distribution. This normalizing transformation is an extension of the Fisher's z-transformation.

AB - Some statistics in common use take the form of a ratio of two statistics, such as sample correlation coefficient, Pearson's coefficient of variation, cumulant estimators, and so on. In this article, using an asymptotic representation of the ratio statistics, we will obtain an Edgeworth expansion and a normalizing transformation with remainder term o(n-1/2). The Edgeworth expansion is based on a Studentized ratio statistic, which is studentized by a consistent variance estimator. Applying these results to the sample correlation coefficient, we obtain the normalizing transformation and an asymptotic confidence interval of the correlation coefficient without assuming specific underlying distribution. This normalizing transformation is an extension of the Fisher's z-transformation.

UR - http://www.scopus.com/inward/record.url?scp=77951982762&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951982762&partnerID=8YFLogxK

U2 - 10.1080/03610920802311741

DO - 10.1080/03610920802311741

M3 - Article

AN - SCOPUS:77951982762

VL - 39

SP - 1344

EP - 1358

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 8-9

ER -