## Abstract

Let Y = (Y_{1}, Y_{2},..., Y_{k})' be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics {I^{λ}, λ ∈ R} introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every k ≥ 4 Pr(2nI ^{λ} (Y) < c) = G_{k-1} (c) + O(n^{-1+μ(k-1)}), where G_{r} (c) is the distribution function of chi-square random variable with r degrees of freedom, μ(r) = 6/(7r + 4) for 3 ≤ r ≤ 7, μ(r) = 5/(6r + 2) for r ≥ 8. This refines Zubov and Ulyanov's result (2008). The proof uses Krä tzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.

Original language | English |
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Pages (from-to) | 115-131 |

Number of pages | 17 |

Journal | Hiroshima Mathematical Journal |

Volume | 40 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2010 |

## Keywords

- Approximation
- Chi-square distribution
- Krätzel-Nowak theorem
- Powerdivergence statistics

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology