Abstract
In this chapter, we focus on approximation problems motivated by studies on the asymptotic behavior of power-divergence family of statistics. These statistics are the goodness-of-fit test statistics and include, in particular, the Pearson chi-squared statistic, the Freeman–Tukey statistic, and the log-likelihood ratio statistic. The distributions of the statistics converge to the chi-squared distribution as sample size \(n\) tends to \(\infty \). We show that the rate of convergence is of order \(n^{-\alpha } \) with \(\alpha : 1/2< \alpha < 1 \). Under some conditions \(\alpha \) is close to 1. The proofs are based on the fundamental number theory results about approximating the number of integer points in convex sets by the Lebesgue measure of the set.
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References
Asylbekov, Zh. A., Zubov, V. N., & Ulyanov, V. V. (2011). On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data. Siberian Mathematical Journal, 52(4), 571–584.
Cressie, N. A. C., & Read, T. R. C. (1984). Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B, 46, 440–464.
Esseen, C. G. (1945). Fourier analysis of distribution functions. Acta Mathematica, 77, 1–125.
Götze, F. (2004). Lattice point problems and values of quadratic forms. Inventiones Mathematicae, 157, 195–226.
Götze, F., &  Ulyanov,  V. V. (2003). Asymptotic disrtribution of \(\chi ^2\)-type statistics, Preprint 03-033, Research group “Spectral analysis, asymptotic distributions and stochastic dynamics”.
Gruber, P. M. (2007). Convex and discrete geometry. New York: Springer.
Hardy, G. (1916). On Dirichlet’s divisor problem. Proceedings of the London Mathematical Society, 15, 1–25.
Hlawka, E. (1950). Über integrale auf konvexen körpern. Monatshefte für Mathematik, 54, 1–36.
Huxley, M. N. (1993). Exponential sums and lattice points II. Proceedings of London Mathematical Society, 66, 279–301.
Pardo, L. (2006). Statistical inference based on divergence measures. Boca Raton: Chapman & Hall/CRC.
Prokhorov, Y. V., & Ulyanov, V. V. (2013). Some approximation problems in statistics and probability. Limit theorems in probability, statistics and number theory (Vol. 42, pp. 235–249)., Springer proceedings in mathematics & statistics Heidelberg: Springer.
Read, T. R. C. (1984). Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics. The Annals of Statistics, 36, 59–69.
Siotani, M., & Fujikoshi, Y. (1984). Asymptotic approximations for the distributions of multinomial goodness-of-fit statistics. Hiroshima Mathematical Journal, 14, 115–124.
Ulyanov, V. V., & Zubov, V. N. (2009). Refinement on the convergence of one family of goodness-of-fit statistics to chi-squared distribution. Hiroshima Mathematical Journal, 39(1), 133–161.
Yarnold, J. K. (1972). Asymptotic approximations for the probability that a sum of lattice random vectors lies in a convex set. The Annals of Mathematical Statistics, 43, 1566–1580.
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Fujikoshi, Y., Ulyanov, V.V. (2020). Power-Divergence Statistics. In: Non-Asymptotic Analysis of Approximations for Multivariate Statistics. SpringerBriefs in Statistics(). Springer, Singapore. https://doi.org/10.1007/978-981-13-2616-5_10
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DOI: https://doi.org/10.1007/978-981-13-2616-5_10
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