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Extreme values, invariance and choice probabilities

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  • Mattsson, Lars-Göran
  • Weibull, Jörgen W.
  • Lindberg, Per Olov

Abstract

Since the pioneering work of McFadden (1974), discrete choice random-utility models have become work horses in many areas in transportation analysis and economics. In these models, the random variables enter additively or multiplicatively and the noise distributions take a particular parametric form. We show that the same qualitative results, with closed-form choice probabilities, can be obtained for a wide class of distributions without such specifications. This class generalizes the statistically independent distributions where any two c.d.f.:s are powers of each others to a class that allows for statistical dependence, in a way analogous to how the independent distributions in the MNL models were generalized into the subclass of MEV distributions that generates the GEV choice models. We show that this generalization is sufficient, and under statistical independence also necessary, for the following invariance property: all conditional random variables, when conditioning upon a certain alternative having been chosen, are identically distributed. While some of these results have been published earlier, we place them in a general unified framework that allows us to extend several of the results and to provide proofs that are simpler, more direct and transparent. Well-known results are obtained as special cases, and we characterize the Gumbel, Fréchet and Weibull distributions.

Suggested Citation

  • Mattsson, Lars-Göran & Weibull, Jörgen W. & Lindberg, Per Olov, 2014. "Extreme values, invariance and choice probabilities," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 81-95.
  • Handle: RePEc:eee:transb:v:59:y:2014:i:c:p:81-95
    DOI: 10.1016/j.trb.2013.10.014
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    References listed on IDEAS

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    5. Hårsman, Björn & Mattsson, Lars-Göran, 2017. "Resolving the entrepreneurship puzzle: Applying Fréchet distributions to Lazear’s occupational choice model," Working Paper Series in Economics and Institutions of Innovation 458, Royal Institute of Technology, CESIS - Centre of Excellence for Science and Innovation Studies, revised 22 Feb 2018.
    6. del Castillo, J.M., 2016. "A class of RUM choice models that includes the model in which the utility has logistic distributed errors," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 1-20.
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    8. Fosgerau, Mogens & Lindberg, Per Olov & Mattsson, Lars-Göran & Weibull, Jörgen, 2015. "Invariance of the distribution of the maximum," MPRA Paper 63538, University Library of Munich, Germany.
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    17. Li, Dawei & Feng, Siqi & Song, Yuchen & Lai, Xinjun & Bekhor, Shlomo, 2023. "Asymmetric closed-form route choice models: Formulations and comparative applications," Transportation Research Part A: Policy and Practice, Elsevier, vol. 171(C).
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