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Empirical Similarity

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Author Info

  • Itzhak Gilboa

    (Tel-Aviv University, HEC, and Yale University)

  • Offer Lieberman

    (The Technion)

  • David Schmeidler

    (Tel-Aviv University and The Ohio State University)

Abstract

An agent is asked to assess a real-valued variable Y p based on certain characteristics X p = (X p-super-1, ..., X p-super-m), and on a database consisting of X i-super-1, ... X i-super-m, Y i) for i = 1, ..., n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Ȳ p-super-s, be the weighted average of all previously observed values Y i, where the weight of Y i for every i = 1, ..., n, is the similarity between the vector X p-super-1, ..., X p-super-m, associated with Y p, and the previously observed vector, X i-super-1, ..., X i-super-m. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations.We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction. Copyright by the President and Fellows of Harvard College and the Massachusetts Institute of Technology.

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File URL: http://www.mitpressjournals.org/doi/pdf/10.1162/rest.88.3.433
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Bibliographic Info

Article provided by MIT Press in its journal The Review of Economics and Statistics.

Volume (Year): 88 (2006)
Issue (Month): 3 (August)
Pages: 433-444

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Handle: RePEc:tpr:restat:v:88:y:2006:i:3:p:433-444

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  1. Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics.
  2. Gabrielle Gayer & Itzhak Gilboa & Offer Lieberman, 2004. "Rule-Based and Case-Based Reasoning in Housing Prices," Cowles Foundation Discussion Papers 1493, Cowles Foundation for Research in Economics, Yale University.
  3. Gilboa, Itzhak & Schmeidler, David, 1995. "Case-Based Decision Theory," The Quarterly Journal of Economics, MIT Press, vol. 110(3), pages 605-39, August.
  4. Billot, Antoine & Gilboa, Itzhak & Schmeidler, David, 2008. "Axiomatization of an exponential similarity function," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 107-115, March.
  5. Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "An Axiomatization of an Exponential Similarity Function," Levine's Bibliography 122247000000000678, UCLA Department of Economics.
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