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Random Choice over a Continuous Set of Options
Stockholm University, Faculty of Science, Department of Mathematics.
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: urn:nbn:se:su:diva-89917OAI: oai:DiVA.org:su-89917DiVA: diva2:621587
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
List of papers
1. Argmax over Continuous Indices of Random Variables - An Approach Using Random Fields
Open this publication in new window or tab >>Argmax over Continuous Indices of Random Variables - An Approach Using Random Fields
(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
Argmax distribution, commuting, extreme value theory, exponential offers, marked point processes, max field.
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-90157 (URN)
Available from: 2013-05-27 Created: 2013-05-27 Last updated: 2013-06-03
2. Extremal Behaviour, Weak Convergence and Argmax Theory for a Class of Non-Stationary Marked Point Processes
Open this publication in new window or tab >>Extremal Behaviour, Weak Convergence and Argmax Theory for a Class of Non-Stationary Marked Point Processes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We formulate a random utility model where we choose from n options1; ; n. The options have associated independent and identicallydistributed (i.i.d) random variables fXi;Uigni=1, where Xi arethe characteristics of option i and Ui is its associated utility.We use the connection between point processes and extreme valuetheory to analyze the statistical properties of choice characteristics Xof the object with the highest utility as n ! 1. We derive analyticexpressions of the asymptotic distribution of choice characteristics fora range of distributional assumptions on the utilities Ui.In our discussion section, we suggest an extension of our method toallow us to further relax our distributional assumptions. We also showhow our theoretical model can be used to explain empirical patternsrelating to commuting time distributions.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-90159 (URN)
Available from: 2013-05-27 Created: 2013-05-27 Last updated: 2013-06-03

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