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Mixtures of Global and Local Edgeworth Expansions and Their Applications

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  • Babu, Gutti Jogesh
  • Bai, Z. D.

Abstract

Edgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive "priors," whenever the third cumulant of the "prior" is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution.

Suggested Citation

  • Babu, Gutti Jogesh & Bai, Z. D., 1996. "Mixtures of Global and Local Edgeworth Expansions and Their Applications," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 282-307, November.
    • Handle: RePEc:eee:jmvana:v:59:y:1996:i:2:p:282-307
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    Cited by:

    1. L. C. Zhao & C. Q. Wu & Q. Wang, 2004. "Berry–Esseen Bound for a Sample Sum from a Finite Set of Independent Random Variables," Journal of Theoretical Probability, Springer, vol. 17(3), pages 557-572, July.
    2. Patrice Bertail & Emilie Chautru & Stephan Clémençon, 2017. "Empirical Processes in Survey Sampling with (Conditional) Poisson Designs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 97-111, March.
    3. Ibrahim Bin Mohamed & Sherzod M. Mirakhmedov, 2016. "Approximation by Normal Distribution for a Sample Sum in Sampling Without Replacement from a Finite Population," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(2), pages 188-220, August.

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