### Abstract

Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by ED^{α} if the variance function is given by μ^{(2-α)/(1-α)}, where μ is the mean function. When 0≤α<1, it is known that the transformation of ED^{(α)} to normality is given by the power transformation X^{(1-2α)/(3-3α)}, and conversely, the power transformation characterizes ED^{(α)}. Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion.

Original language | English |
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Pages (from-to) | 173-186 |

Number of pages | 14 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 45 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 1993 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

**Convergence of the Gram-Charier expansion after the normalizing Box-Cox transformation.** / Nishii, Ryuei.

Research output: Contribution to journal › Article

*Annals of the Institute of Statistical Mathematics*, vol. 45, no. 1, pp. 173-186. https://doi.org/10.1007/BF00773677

}

TY - JOUR

T1 - Convergence of the Gram-Charier expansion after the normalizing Box-Cox transformation

AU - Nishii, Ryuei

PY - 1993/3/1

Y1 - 1993/3/1

N2 - Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by EDα if the variance function is given by μ(2-α)/(1-α), where μ is the mean function. When 0≤α<1, it is known that the transformation of ED(α) to normality is given by the power transformation X(1-2α)/(3-3α), and conversely, the power transformation characterizes ED(α). Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion.

AB - Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by EDα if the variance function is given by μ(2-α)/(1-α), where μ is the mean function. When 0≤α<1, it is known that the transformation of ED(α) to normality is given by the power transformation X(1-2α)/(3-3α), and conversely, the power transformation characterizes ED(α). Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion.

UR - http://www.scopus.com/inward/record.url?scp=0006342776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0006342776&partnerID=8YFLogxK

U2 - 10.1007/BF00773677

DO - 10.1007/BF00773677

M3 - Article

AN - SCOPUS:0006342776

VL - 45

SP - 173

EP - 186

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 1

ER -