A maximum likelihood approach to inference under coarse data based on minimax regret

Abstract

Various 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: maximize the maximal likelihood induced by precise datasets compatible with the incomplete observations, or maximize the minimal such likelihood. These approaches prove to be extremist, 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 minimax relative regret criterion with respect to maximal likelihood solutions obtained for all precise datasets compatible with the coarse data. It uses relative likelihood and seems to achieve a trade-off between the maximax and the maximin methods.