A Nonparametric Distance Model for Unidimensional Unfolding

Abstract

The unidimensional unfolding model is placed in the wider context of social choice theory, median procedures and strictly unimodal distance models for rankings. Social choice theory is used to construct a framework for the unfolding model; for example, given single-peaked preference functions for individuals, Simple Majority Rule yields the median ordering as a group consensus ordering. We generalize Coombs’ and Goodman’s (1954) theorems: if the data follow a strictly unimodal distance model, the median ordering is an admissible ordering of the J scale that has the highest probability. This is because the maximum likelihood and the minimum-number-of-inversions criterion yield the same ordering in a strictly unimodal distance model: the mean/modal/median ordering. We prove that the group consensus ordering is transitive and is the modal or median ordering. Also, we prove that the social preference function is unimodal on the J scale in this case.Key words and phrasesunfoldingmedian orderinggroup consensus rankingKemeny distanceunimodal distance model for rankingsmaximum likelihoodminimum number of inversions