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On the Optimality of Stein Factors

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Probability Approximations and Beyond

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 205))

Abstract

The application of Stein’s method for distributional approximation often involves so-called Stein factors (also called magic factors) in the bound of the solutions to Stein equations. However, in some cases these factors contain additional (undesirable) logarithmic terms. It has been shown for many Stein factors that the known bounds are sharp and thus that these additional logarithmic terms cannot be avoided in general. However, no probabilistic examples have appeared in the literature that would show that these terms in the Stein factors are not just unavoidable artefacts, but that they are there for a good reason. In this article we close this gap by constructing such examples. This also leads to a new interpretation of the solutions to Stein equations.

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Acknowledgments

The author would like to thank Gesine Reinert and Dominic Schuhmacher for fruitful discussions and an anonymous referee for helpful comments.

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Authors and Affiliations

  1. Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore, 117546, Republic of Singapore

    Adrian Röllin

Authors
  1. Adrian Röllin
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Correspondence to Adrian Röllin .

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Editors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, Zürich, 8057, Switzerland

    Andrew Barbour

  2. , Department of Statistics and Applied Pro, National University of Singapore, Singapore, 119260, Singapore

    Hock Peng Chan

  3. Dept. Statistics, Stanford University, Serra Mall 390, Stanford, 94305, California, USA

    David Siegmund

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Röllin, A. (2012). On the Optimality of Stein Factors. In: Barbour, A., Chan, H., Siegmund, D. (eds) Probability Approximations and Beyond. Lecture Notes in Statistics(), vol 205. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1966-2_5

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