Abstract
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) <c) = Gk-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.
| 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.
In: Hiroshima Mathematical Journal, Vol. 40, No. 1, 03.2010, p. 115-131.Research output: Contribution to journal › Article
}
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 -