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On Hodges and Lehmann’s “6/π Result”

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Contemporary Developments in Statistical Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 68))

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Abstract

While the asymptotic relative efficiency (ARE) of Wilcoxon rank-based tests for location and regression with respect to their parametric Student competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that the ARE of the same Wilcoxon tests with respect to their van der Waerden or normal-score counterparts is bounded from above by \(6/\pi\approx 1.910\). In this chapter, we revisit that result, and investigate similar bounds for statistics based on Student scores. We also consider the serial version of this ARE. More precisely, we study the ARE, under various densities, of the Spearman–Wald–Wolfowitz and Kendall rank-based autocorrelations with respect to the van der Waerden or normal-score ones used to test (ARMA) serial dependence alternatives.

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Acknowledgement

This note originates in a research visit by the last two authors to the Department of Operations Research and Financial Engineering (ORFE) at Princeton University in the Fall of 2012; ORFE’s support and hospitality are gratefully acknowledged. Marc Hallin’s research is supported by the Sonderforschungsbereich “Statistical modelling of nonlinear dynamic processes” (SFB 823) of the Deutsche Forschungsgemeinschaft, a Discovery Grant of the Australian Research Council, and the IAP research network grant P7/06 of the Belgian government (Belgian Science Policy). We gratefully acknowledge the pertinent comments by an anonymous referee on the original version of the manuscript, which lead to substantial improvements.

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Authors and Affiliations

  1. ECARES, Université libre de Bruxelles, CP114/4, 50 Ave. F.D. Roosevelt B-1050, Bruxelles, Belgium

    Marc Hallin

  2. ORFE, Princeton University, Sherrerd Hall, 08544, Princeton, NJ, USA

    Marc Hallin

  3. Faculté des Sciences, de la Technologie et de la Communication, Université de Luxembourg, Campus Kirchberg, Mathematics Research Unit, BLG, 6 rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Grand Duchy of Luxembourg

    Yvik Swan

  4. EQUIPPE and INRIA, Université Lille 3, Domaine universitaire du Pont de Bois, rue du Barreau BP 60149, 59653, Villeneuve-d’Ascq Cedex, France

    Thomas Verdebout

Authors
  1. Marc Hallin
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  2. Yvik Swan
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  3. Thomas Verdebout
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Correspondence to Marc Hallin .

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Editors and Affiliations

  1. North Carolina State Universy, Raleigh, North Carolina, USA

    Soumendra Lahiri

  2. Department of Mathematical Sciences, Binghampton University, Binghamton, New York, USA

    Anton Schick

  3. Applied Statistics Unit, Kolkata, India

    Ashis SenGupta

  4. Department of Statistics, University of Georgia, Athens, Georgia, USA

    T.N. Sriram

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Hallin, M., Swan, Y., Verdebout, T. (2014). On Hodges and Lehmann’s “6/π Result”. In: Lahiri, S., Schick, A., SenGupta, A., Sriram, T. (eds) Contemporary Developments in Statistical Theory. Springer Proceedings in Mathematics & Statistics, vol 68. Springer, Cham. https://doi.org/10.1007/978-3-319-02651-0_9

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