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Argmax over Continuous Indices of Random Variables - An Approach Using Random Fields
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

optimizationover a discrete number of random variables. In this paperwe extend this theory from the discrete to the continuous case, andconsider the limiting distribution of the location of the best offer asthe number of offers tends to infinity.Given a set   Rd of possible offers we seek a distribution over ,the argmax measure of the best offer. It depends on , the samplingdistribution of offer locations, and a measure index , which assignsto each point x 2  a probability distribution of offers.This problem is closely related to argmax theory of marked pointprocesses, altough we consider deterministic sequences of points inspace, to allow for greater generality. We first define a finite sampleargmax measure and then give conditions under which it converges asthe number of offers tends to infinity.To this end, we introduce a max-field of best offers and use continuityproperties of this field to calculate the argmax measure. Wedemonstrate the usefulness of the method by giving explicit formulasfor the limiting argmax distribution for a large class of models, includingexponential independent offers with a deterministic, additivedisturbance term. Finally, we illustrate the theory by simulations.

Keyword [en]
Argmax distribution, commuting, extreme value theory, exponential offers, marked point processes, max field.
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:su:diva-90157OAI: oai:DiVA.org:su-90157DiVA: diva2:623284
Available from: 2013-05-27 Created: 2013-05-27 Last updated: 2013-06-03
In thesis
1. Random Choice over a Continuous Set of Options
Open this publication in new window or tab >>Random Choice over a Continuous Set of Options
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Random choice theory has traditionally modeled choices over a -nite number of options. This thesis generalizes the literature by studyingthe limiting behavior of choice models as the number of optionsapproach a continuum.The thesis uses the theory of random elds, extreme value theoryand point processes to calculate this limiting behavior. For a numberof distributional assumptions, we can give analytic expressions forthe limiting probability distribution of the characteristics of the bestchoice. In addition, we also outline a straightforward extension to ourtheory which would signicantly relax the distributional assumptionsneeded to derive analytical results.Some examples from commuting research are discussed to illustratepotential applications of the theory.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2013. 65 p.
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-89917 (URN)
Presentation
2013-06-05, Rum 5:306 Matematiska Institutionen, Kräftriket, Stockholm, 15:00 (English)
Supervisors
Available from: 2013-05-22 Created: 2013-05-15 Last updated: 2013-06-03Bibliographically approved

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