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

Pages (from-to) | 115-131 |

Number of pages | 17 |

Journal | Hiroshima Mathematical Journal |

Volume | 40 |

Issue number | 1 |

Publication status | Published - Mar 2010 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology

### Cite this

**Convergence rate of multinomial goodness-of-fit statistics to chi-square distribution.** / Zhenisbek, Assylbekov.

Research output: Contribution to journal › Article

*Hiroshima Mathematical Journal*, vol. 40, no. 1, pp. 115-131.

}

TY - JOUR

T1 - Convergence rate of multinomial goodness-of-fit statistics to chi-square distribution

AU - Zhenisbek, Assylbekov

PY - 2010/3

Y1 - 2010/3

N2 - Let Y = (Y1, Y2,..., Yk)' be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics {Iλ, λ ∈ R} introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every k ≥ 4 Pr(2nI λ (Y) k-1 (c) + O(n-1+μ(k-1)), where Gr (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.

AB - Let Y = (Y1, Y2,..., Yk)' be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics {Iλ, λ ∈ R} introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every k ≥ 4 Pr(2nI λ (Y) k-1 (c) + O(n-1+μ(k-1)), where Gr (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.

KW - Approximation

KW - Chi-square distribution

KW - Krätzel-Nowak theorem

KW - Powerdivergence statistics

UR - http://www.scopus.com/inward/record.url?scp=77952700609&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952700609&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77952700609

VL - 40

SP - 115

EP - 131

JO - Hiroshima Mathematical Journal

JF - Hiroshima Mathematical Journal

SN - 0018-2079

IS - 1

ER -