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Annals of the Institute of Statistical Mathematics

, Volume 38, Issue 2, pp 205–222

# Modified information criteria for a uniform approximate equivalence of probability distributions

• T Matsunawa
Article
• 11 Downloads

## Summary

Information criteria for two-sided uniform ϕ-equivalence, which is a newly introduced strong approximate equivalence of probability distributions, are proposed. The criteria resort to some modified K-L informations defined on suitable approximate main domains and are presented in the form of systems with double inequalities. They present systematic implements to handle many statistical approximation problems and are useful to evaluate related approximation errors quantitatively. Criteria for asymptotic cases are also derived from the presented inequalities. As applications, necessary and sufficient conditions and error evaluations are given for approximate and/or asymptotic equivalences of the probability distributions on sampling with and without replacement from a finite population and on quasi-extreme order statistics from a continuous distribution.

## Key words

Modified information criteria K-L information uniform ϕ-equivalence sampling from finite population quasi-extreme order statistics

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## References

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Freedman, D. (1977) A remak on the difference between sampling with and without replacement,J. Amer. Statist. Ass.,72, 681.
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Ikeda, S. and Matsunawa, T. (1985). A necessary and sufficient condition for the uniform asymptotic equivalence with application,Statistical Theory and Data Analysis, (ed. K. Matusita), North-Holland, 257–268.Google Scholar
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Matsunawa, T. and Ikeda, S. (1976). Uniform asymptotic distribution of extremes,Essays in Probability and Statistics, (eds. S. Ikeda et al.), Shinko Tsusho Co. Ltd., 419–432.Google Scholar
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Matsunawa, T. (1976). Some inequalities based on inverse factorial series,Ann. Inst. Statist. Math.,28, 291–305.
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Matsunawa, T. (1982). Uniform ϕ-equivalence of probability distributions based on information and related measures of discrepancy,Ann. Inst. Statist. Math.,34, A, 1–17.

## Copyright information

© The Institute of Statistical Mathematics, Tokyo 1986

## Authors and Affiliations

• T Matsunawa

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