### Abstract

We study the rate of weak convergence of the distributions of the statistics {t_{λ}(Y), λ ∈ ℝ} from the power divergence family of statistics to the χ^{2} distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that, where G_{2}(c) is the χ^{2} distribution function of a random variable with two degrees of freedom. In the proof we use Huxley's theorem of 1993 on approximating the number of integer points in a plane convex set with smooth boundary by the area of the set.

Original language | English |
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Pages (from-to) | 571-584 |

Number of pages | 14 |

Journal | Siberian Mathematical Journal |

Volume | 52 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- accuracy of χ approximation
- Huxley theorem
- integer points
- power divergence family of statistics

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Siberian Mathematical Journal*,

*52*(4), 571-584. https://doi.org/10.1134/S0037446611040021

**On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data.** / Asylbekov, Zh A.; Zubov, V. N.; Ulyanov, V. V.

Research output: Contribution to journal › Article

*Siberian Mathematical Journal*, vol. 52, no. 4, pp. 571-584. https://doi.org/10.1134/S0037446611040021

}

TY - JOUR

T1 - On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data

AU - Asylbekov, Zh A.

AU - Zubov, V. N.

AU - Ulyanov, V. V.

PY - 2011/7

Y1 - 2011/7

N2 - We study the rate of weak convergence of the distributions of the statistics {tλ(Y), λ ∈ ℝ} from the power divergence family of statistics to the χ2 distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that, where G2(c) is the χ2 distribution function of a random variable with two degrees of freedom. In the proof we use Huxley's theorem of 1993 on approximating the number of integer points in a plane convex set with smooth boundary by the area of the set.

AB - We study the rate of weak convergence of the distributions of the statistics {tλ(Y), λ ∈ ℝ} from the power divergence family of statistics to the χ2 distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that, where G2(c) is the χ2 distribution function of a random variable with two degrees of freedom. In the proof we use Huxley's theorem of 1993 on approximating the number of integer points in a plane convex set with smooth boundary by the area of the set.

KW - accuracy of χ approximation

KW - Huxley theorem

KW - integer points

KW - power divergence family of statistics

UR - http://www.scopus.com/inward/record.url?scp=80051991971&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80051991971&partnerID=8YFLogxK

U2 - 10.1134/S0037446611040021

DO - 10.1134/S0037446611040021

M3 - Article

VL - 52

SP - 571

EP - 584

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 4

ER -