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Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications

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  • Marc Hallin
  • Jean-Marie Dufour

Abstract

The problem of evaluating tail probabilities for linear combinations of independent, possibly nonidentically distributed, bounded random variables arises in various statistical contexts, mainly connected with nonparametric inference. A remarkable inequality on such tail probabilities has been established by Eaton. The significance of Eaton’s inequality is substantiated by a recent result of Pinelis showing that the minimum BEPof Eaton’s bound BEand a traditional Chebyshev bound yields an inequality that is optimal within a fairly general class of bounds. Eaton’s bound, however, is not directly operational, because it is not explicit; apparently, it never has been studied numerically, and its many potential statistical applications have not yet been considered. A simpler inequality recently proposed by Edelman for linear combinations of iid Bernoulli variables is also considered, but it appears considerably less tight than Eaton’s original bound. This article has three main objectives. First, we put Eaton’s exact bound BEinto numerically tractable form and tabulate it, along with BEP, which makes them readily applicable; the resulting conservative critical values are provided for standard significance levels. Second, we show how further improvement can be obtained over the Eaton-Pinelis bound BEPif the number n of independent variables in the linear combination under study is taken into account. The resulting improved Eaton bounds B+EPand the corresponding conservative critical values are also tabulated for standard significance levels and most empirically relevant values of n. Finally, various statistical applications are discussed: permutation t tests against location shifts, permutation t tests against regression or trend, permutation tests against serial correlation, and linear signed rank tests against various alternatives, all in the presence of possibly nonidentically distributed (e.g. heteroscedastic) data. For permutation t tests and linear signed rank tests, the improved Eaton bounds are compared numerically with other available bounds. The results indicate that the sharpened Eaton bounds often yield sizable improvements over all other bounds considered. © 1993 Taylor & Francis Group, LLC.

Suggested Citation

  • Marc Hallin & Jean-Marie Dufour, 1993. "Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications," ULB Institutional Repository 2013/2043, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:ulb:ulbeco:2013/2043 Note: SCOPUS: ar.j
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    Cited by:

    1. Dufour, Jean-Marie & Farhat, Abdeljelil & Hallin, Marc, 2006. "Distribution-free bounds for serial correlation coefficients in heteroskedastic symmetric time series," Journal of Econometrics, Elsevier, vol. 130(1), pages 123-142, January.
    2. Gossner, Olivier & Schlag, Karl H., 2013. "Finite-sample exact tests for linear regressions with bounded dependent variables," Journal of Econometrics, Elsevier, vol. 177(1), pages 75-84.
    3. Iosif Pinelis, 2014. "An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality," Risks, MDPI, Open Access Journal, vol. 2(3), pages 1-44, September.
    4. Iosif Pinelis, 2013. "An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality," Papers 1310.6025, arXiv.org.
    5. Oliver Gossner & Karl Schlag, 2012. "Finite Sample Exact tests for Linear," Vienna Economics Papers 1201, University of Vienna, Department of Economics.
    6. Donald Brown & Rustam Ibragimov, 2005. "Sign Tests for Dependent Observations and Bounds for Path-Dependent Options," Yale School of Management Working Papers amz2581, Yale School of Management, revised 01 Jul 2005.
    7. Karl H.Schlag, 2015. "Who gives Direction to Statistical Testing? Best Practice meets Mathematically Correct Tests," Vienna Economics Papers 1512, University of Vienna, Department of Economics.
    8. Pinelis, Iosif, 2013. "An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality," MPRA Paper 51361, University Library of Munich, Germany.
    9. Donald J. Brown & Rustam Ibragimov, 2005. "Sign Tests for Dependent Observations and Bounds for Path-Dependent Options," Cowles Foundation Discussion Papers 1518, Cowles Foundation for Research in Economics, Yale University.
    10. Karl Schlag & Olivier Gossner, 2010. "Finite sample nonparametric tests for linear regressions," Economics Working Papers 1212, Department of Economics and Business, Universitat Pompeu Fabra.

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