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On stationary multiplier methods for the rounding of probabilities and the limiting law of the Sainte-Laguë divergence

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  • Heinrich Lothar
  • Pukelsheim Friedrich
  • Schwingenschlögl Udo

Abstract

Stationary multiplier methods are procedures for rounding real probabilities into rational proportions, while the Sainte-Laguë divergence is a reasonable measure for the cumulative error resulting from this rounding step. Assuming the given probabilities to be uniformly distributed, we show that the Sainte-Laguë divergences converge to the Lévy-stable distribution that obtains for the multiplier method with standard rounding. The norming constants to achieve convergence depend in a subtle way on the stationary method used.

Suggested Citation

  • Heinrich Lothar & Pukelsheim Friedrich & Schwingenschlögl Udo, 2005. "On stationary multiplier methods for the rounding of probabilities and the limiting law of the Sainte-Laguë divergence," Statistics & Risk Modeling, De Gruyter, vol. 23(2), pages 117-129, February.
    • Handle: RePEc:bpj:strimo:v:23:y:2005:i:2/2005:p:117-129:n:2
    DOI: 10.1524/stnd.2005.23.2.117
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    References listed on IDEAS

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    1. Udo Schwingenschlögl & Mathias Drton, 2004. "Seat allocation distributions and seat biases of stationary apportionment methods for proportional representation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 60(2), pages 191-202, September.
    2. Mathias Drton & Udo Schwingenschlögl, 2005. "Asymptotic seat bias formulas," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 62(1), pages 23-31, September.
    3. 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.
    4. 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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    Cited by:

    1. Grimmett, G.R. & Oelbermann, K.-F. & Pukelsheim, F., 2012. "A power-weighted variant of the EU27 Cambridge Compromise," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 136-140.
    2. Udo Schwingenschlögl & Friedrich Pukelsheim, 2006. "Seat Biases in Proportional Representation Systems with Thresholds," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(1), pages 189-193, August.
    3. Niemeyer, Horst F. & Niemeyer, Alice C., 2008. "Apportionment methods," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 240-253, September.

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