A min-max regret approach to maximum likelihood inference under incomplete data

International audienceVarious methods have been proposed to express and solve maximum likelihood problems with incomplete data. In some of these approaches, the idea is that incompleteness makes the likelihood function imprecise. Two proposals can be found to cope with this situation: the maximax approach that maximizes the greatest likelihood value induced by precise data sets compatible with the incomplete observations, and the maximin approach that maximizes the least such likelihood value. These approaches prove to display extreme behaviors in some contexts, the maximax approach having a tendency to disambiguate the data, while the maximin approach favors uniform distributions. In this paper, we propose an alternative approach consisting in minimizing a relative regret criterion with respect to maximal likelihood solutions obtained for all precise data sets compatible with the coarse data. In contrast with the maximax and the maximin methods, the min-max-regret method relies on comparing relative likelihoods and obtains results that achieve a trade-off between results of the two other methods. The methods are compared on toy examples and also on simulated random data as well as a supervised classification problem

Modeling random and non-random decision uncertainty in ratings data: A fuzzy beta model

Modeling human ratings data subject to raters' decision uncertainty is an attractive problem in applied statistics. In view of the complex interplay between emotion and decision making in rating processes, final raters' choices seldom reflect the true underlying raters' responses. Rather, they are imprecisely observed in the sense that they are subject to a non-random component of uncertainty, namely the decision uncertainty. The purpose of this article is to illustrate a statistical approach to analyse ratings data which integrates both random and non-random components of the rating process. In particular, beta fuzzy numbers are used to model raters' non-random decision uncertainty and a variable dispersion beta linear model is instead adopted to model the random counterpart of rating responses. The main idea is to quantify characteristics of latent and non-fuzzy rating responses by means of random observations subject to fuzziness. To do so, a fuzzy version of the Expectation-Maximization algorithm is adopted to both estimate model's parameters and compute their standard errors. Finally, the characteristics of the proposed fuzzy beta model are investigated by means of a simulation study as well as two case studies from behavioral and social contexts.Comment: 24 pages, 0 figures, 5 table