IDEAS home Printed from https://ideas.repec.org/a/bpj/strimo/v23y2005i2-2005p117-129n2.html
   My bibliography  Save this article

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

Author

Listed:
  • 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(2/2005), 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
    as

    Download full text from publisher

    File URL: https://doi.org/10.1524/stnd.2005.23.2.117
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1524/stnd.2005.23.2.117?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Schwingenschlögl, Udo, 2007. "Probabilities of majority and minority violation in proportional representation," Statistics & Probability Letters, Elsevier, vol. 77(17), pages 1690-1695, November.
    2. Luc Lauwers & Tom Van Puyenbroeck, 2006. "The Hamilton Apportionment Method Is Between the Adams Method and the Jefferson Method," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 390-397, May.
    3. Svante Janson, 2014. "Asymptotic bias of some election methods," Annals of Operations Research, Springer, vol. 215(1), pages 89-136, April.
    4. Schwingenschlögl, Udo & Drton, Mathias, 2006. "Seat excess variances of apportionment methods for proportional representation," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1723-1730, October.
    5. Jarosław Flis & Wojciech Słomczyński & Dariusz Stolicki, 2020. "Pot and ladle: a formula for estimating the distribution of seats under the Jefferson–D’Hondt method," Public Choice, Springer, vol. 182(1), pages 201-227, January.
    6. Udo Schwingenschlögl, 2008. "Asymptotic Equivalence of Seat Bias Models," Statistical Papers, Springer, vol. 49(2), pages 191-200, April.
    7. Laszlo A. Koczy & Peter Biro & Balazs Sziklai, 2017. "US vs. European Apportionment Practices: The Conflict between Monotonicity and Proportionality," CERS-IE WORKING PAPERS 1716, Institute of Economics, Centre for Economic and Regional Studies.
    8. 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.
    9. Laszlo A. Koczy & Balazs Sziklai, 2018. "Bounds on Malapportionment," CERS-IE WORKING PAPERS 1801, Institute of Economics, Centre for Economic and Regional Studies.
    10. Bittó, Virág, 2017. "Az Imperiali és Macau politikai választókörzet-kiosztási módszerek empirikus vizsgálata [Empirical Analysis of the Imperiali and Macau Apportionment Methods]," MPRA Paper 79554, University Library of Munich, Germany.
    11. Steven J Brams & D Marc Kilgour, 2012. "Narrowing the field in elections: The Next-Two rule," Journal of Theoretical Politics, , vol. 24(4), pages 507-525, October.
    12. Brams,S.L. & Kaplan,T.R., 2002. "Dividing the indivisible : procedures for allocating cabinet ministries to political parties in a parliamentary system," Center for Mathematical Economics Working Papers 340, Center for Mathematical Economics, Bielefeld University.
    13. Ulrich Kohler & Janina Zeh, 2012. "Apportionment methods," Stata Journal, StataCorp LP, vol. 12(3), pages 375-392, September.
    14. Balázs R Sziklai & Károly Héberger, 2020. "Apportionment and districting by Sum of Ranking Differences," PLOS ONE, Public Library of Science, vol. 15(3), pages 1-20, March.
    15. 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(1/2004), pages 43-60, January.
    16. Tom Van Puyenbroeck, 2008. "Proportional Representation, Gini Coefficients, and the Principle of Transfers," Journal of Theoretical Politics, , vol. 20(4), pages 498-526, October.
    17. Jones, Michael A. & McCune, David & Wilson, Jennifer M., 2020. "New quota-based apportionment methods: The allocation of delegates in the Republican Presidential Primary," Mathematical Social Sciences, Elsevier, vol. 108(C), pages 122-137.
    18. Paul Edelman, 2015. "Voting power apportionments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(4), pages 911-925, April.
    19. Jones, Michael A. & Wilson, Jennifer M., 2010. "Evaluation of thresholds for power mean-based and other divisor methods of apportionment," Mathematical Social Sciences, Elsevier, vol. 59(3), pages 343-348, May.
    20. Słomczyński, Wojciech & Życzkowski, Karol, 2012. "Mathematical aspects of degressive proportionality," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 94-101.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bpj:strimo:v:23:y:2005:i:2/2005:p:117-129:n:2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Peter Golla (email available below). General contact details of provider: https://www.degruyter.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.