Prof. Michael Smithson,
Australian National University, Australia
Univariate and Multivariate Distributions for Fuzzy Membership Data
Fuzzy data can take many different forms, but perhaps the most common kind include fuzzy membership values on the closed unit interval, [0,1]. Appropriate statistical analyses of such data treat them as random variables on the unit interval, which motivates the topic for this talk. A survey of distributions for random variables on the unit interval reveals several practical problems and (in some instances) their solutions:
- Nearly all distributions have defined densities only on the open interval (0,1), thereby excluding 0’s and 1’s. There are some remedies for this, although they are not always satisfactory.
- Until recently, there was a very limited variety of available distributions for unit interval variables, especially for modelling their quantiles. However, this class of distributions has been expanded considerably.
- The most well-known distributions for unit interval variables have two parameters. Introducing more parameters often results in severe collinearity among parameter estimates. This still is an outstanding problem, but mixture distributions may be a way out.
- Few multivariate distribution models are available for collections of unit interval variables, and the popular Dirichlet distribution is limited to compositional data. Copulas may provide a partial solution to this problem.
Dr. Michael Smithson
Research School of Psychology, Bldg 39, room 215
The Australian National University
Canberra, A.C.T. 2601 AUSTRALIA
Michael Smithson is a Professor in the Research School of Psychology at The Australian National University and received his PhD from the University of Oregon. He is the author of Confidence Intervals (2003), Statistics with Confidence (2000), Ignorance and Uncertainty (1989), and Fuzzy Set Analysis for the Behavioral and Social Sciences (1987); co-author of Fuzzy Set Theory: Applications in the Social Sciences (2006), Generalized Linear Models for Categorical and Limited Dependent Variables (2014), and Generalized Linear Models for Bounded and Limited Quantitative Variables (2020); and co-editor of Uncertainty and Risk: Multidisciplinary Perspectives (2008) and Resolving Social Dilemmas: Dynamic, Structural, and Intergroup Aspects (1999). His other publications include more than 180 refereed journal articles and book chapters. His primary research interests are in judgment and decision making under ignorance and uncertainty, statistical methods for the social sciences, and applications of fuzzy set theory to the social sciences.