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Exploring the relation between the r* approximation and the Edgeworth expansion

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  • Jorge Arevalillo

Abstract

In this paper we study the relation between the r* saddlepoint approximation and the Edgeworth expansion when quite general assumptions for the statistic under consideration are fulfilled. We will show that the two term Edgeworth expansion approximates the r* formula up to an O(n −3/2 ) remainder; this provides a new way of looking at the order of the error of the r* approximation. This finding will be used to inspect the close connection between the r* formula and the Edgeworth B adjustment introduced in Phillips (Biometrika 65:91–98, 1978 ). We will show that, whenever an Edgeworth expansion exists, this adjustment approximates both the distribution function of the statistic and the r* formula to the same order degree as the Edgeworth expansion. Some numerical examples for the sample mean and U-statistics are given in order to shed light on the theoretical discussion. Copyright Springer-Verlag 2012

Suggested Citation

  • Jorge Arevalillo, 2012. "Exploring the relation between the r* approximation and the Edgeworth expansion," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1009-1024, November.
    • Handle: RePEc:spr:metrik:v:75:y:2012:i:8:p:1009-1024
    DOI: 10.1007/s00184-011-0365-5
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    References listed on IDEAS

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    1. Killmann Frank & von Collani Elart, 2001. "A Note on the Convolution of the Uniform and Related Distributions and Their Use in Quality Control," Stochastics and Quality Control, De Gruyter, vol. 16(1), pages 17-41, January.
    2. Yoshimichi Ochi, 1983. "Asymptotic Expansions For The Distribution Of An Estimator In The First‐Order Autoregressive Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(1), pages 57-67, January.
    3. Monti, Anna Clara, 1993. "A new look at the relationship between Edgeworth expansion and saddlepoint approximation," Statistics & Probability Letters, Elsevier, vol. 17(1), pages 49-52, May.
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    Cited by:

    1. Arevalillo, Jorge M, 2024. "On the empirical approximation to quantiles from Lugannani–Rice saddlepoint formula," Statistics & Probability Letters, Elsevier, vol. 209(C).
    2. Jorge Arevalillo, 2014. "Higher-order approximations to the quantile of the distribution for a class of statistics in the first-order autoregression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 291-310, June.

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