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Sainte-Laguë’s chi-square divergence for the rounding of probabilities and its convergence to a stable law

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  • Heinrich Lothar
  • Pukelsheim Friedrich
  • Schwingenschlögl Udo

Abstract

For rounding arbitrary probabilities on finitely many categories to rational proportions, the multiplier method with standard rounding stands out. Sainte-Laguë showed in 1910 that the method minimizes a goodness-of-fit criterion that nowadays classifies as a chi-square divergence. Assuming the given probabilities to be uniformly distributed, we derive the limiting law of the Sainte-Laguë divergence, first when the rounding accuracy increases, and then when the number of categories grows large. The latter limit turns out to be a Lévy-stable distribution.

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  • Heinrich Lothar & Pukelsheim Friedrich & Schwingenschlögl Udo, 2004. "Sainte-Laguë’s chi-square divergence for the rounding of probabilities and its convergence to a stable law," Statistics & Risk Modeling, De Gruyter, vol. 22(1), pages 43-60, January.
    • Handle: RePEc:bpj:strimo:v:22:y:2004:i:1/2004:p:43-60:n:4
    DOI: 10.1524/stnd.22.1.43.32717
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    References listed on IDEAS

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    1. Friedrich Pukelsheim & Albert W. Marshall & Ingram Olkin, 2002. "A majorization comparison of apportionment methods in proportional representation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 885-900.
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    Cited by:

    1. Lothar Heinrich & Udo Schwingenschlögl, 2006. "Goodness-of-fit Criteria for the Adams and Jefferson Rounding Methods and their Limiting Laws," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 64(2), pages 191-207, October.

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