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Asymptotic Approximations for the Distributions of the Multinomial Goodness-of-Fit Statistics under Local Alternatives

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  • Taneichi, Nobuhiro
  • Sekiya, Yuri
  • Suzukawa, Akio

Abstract

N. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B46, 440-464) introduced a class of multinomial goodness-of-fit statistics Ra based on power divergence. All Ra have the same chi-square limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of Ra under local alternatives. We obtain an expression of approximation for the distribution of Ra under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of Ra. We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of Ra. We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance.

Suggested Citation

  • Taneichi, Nobuhiro & Sekiya, Yuri & Suzukawa, Akio, 2002. "Asymptotic Approximations for the Distributions of the Multinomial Goodness-of-Fit Statistics under Local Alternatives," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 335-359, May.
    • Handle: RePEc:eee:jmvana:v:81:y:2002:i:2:p:335-359
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    References listed on IDEAS

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    1. Sadao Tomizawa & Takashi Seo & Hideharu Yamamoto, 1998. "Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories," Journal of Applied Statistics, Taylor & Francis Journals, vol. 25(3), pages 387-398.
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    Cited by:

    1. Bogatyrev, S.A. & Götze, F. & Ulyanov, V.V., 2006. "Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 2041-2056, October.
    2. Taneichi, Nobuhiro & Sekiya, Yuri & Toyama, Jun, 2019. "Transformed statistics for tests of conditional independence in J×K×L contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 193-208.
    3. Taneichi, Nobuhiro & Sekiya, Yuri, 2007. "Improved transformed statistics for the test of independence in rxs contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1630-1657, September.
    4. Sekiya, Yuri & Taneichi, Nobuhiro, 2004. "Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 199-223, November.
    5. Taneichi, Nobuhiro & Sekiya, Yuri & Toyama, Jun, 2014. "Transformed goodness-of-fit statistics for a generalized linear model of binary data," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 311-329.
    6. Taneichi, Nobuhiro & Sekiya, Yuri & Toyama, Jun, 2011. "Improved transformed deviance statistic for testing a logistic regression model," Journal of Multivariate Analysis, Elsevier, vol. 102(9), pages 1263-1279, October.

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