Poisson and Gaussian approximations of the power divergence family of statistics

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Abstract

Consider the family of power divergence statistics based on n trials, each leading to one of r possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when r is of order n2 and the allocation is asymptotically uniform as n→∞) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of n, r and the index parameter λ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics.
Original languageEnglish
Pages (from-to)25-37
Number of pages13
JournalALEA: Latin American Journal of Probability and Mathematical Statistics
Volume21
DOIs
Publication statusPublished - Jan 2024

Keywords

  • Cressie-Read statistics
  • Pearson's statistic
  • Poisson approximation
  • log-likelihood ratio
  • normal approximation
  • uniform metric

ASJC Scopus subject areas

  • Statistics and Probability

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