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On Sharp Identification Regions for Regression Under Interval Data

The reliable analysis of interval data (coarsened data) is one of the 
most promising applications of imprecise probabilities in statistics. If one 
refrains from making untestable, and often materially unjustified, strong 
assumptions on the coarsening process, then the empirical distribution 
of the data is imprecise, and statistical models are, in Manski’s terms, 
partially identified. We first elaborate some subtle differences between 
two natural ways of handling interval data in the dependent variable of 
regression models, distinguishing between two different types of identification 
regions, called Sharp Marrow Region (SMR) and Sharp Collection 
Region (SCR) here. Focusing on the case of linear regression analysis, we 
then derive some fundamental geometrical properties of SMR and SCR, 
allowing a comparison of the regions and providing some guidelines for 
their canonical construction. 
Relying on the algebraic framework of adjunctions of two mappings between 
partially ordered sets, we characterize SMR as a right adjoint and 
as the monotone kernel of a criterion function based mapping, while SCR 
is indeed interpretable as the corresponding monotone hull. Finally we 
sketch some ideas on a compromise between SMR and SCR based on a 
set-domained loss function. 
This paper is an extended version of a shorter paper with the same title, 
that is conditionally accepted for publication in the Proceedings of 
the Eighth International Symposium on Imprecise Probability: Theories 
and Applications. In the present paper we added proofs and the seventh 
chapter with a small Monte-Carlo-Illustration, that would have made the 
original paper too long.