Georg Schollmeyer, Thomas Augustin

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On Sharp Identification Regions for Regression Under Interval Data

Abstract

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 \em 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.

Keywords

partial identification, imprecise probabilities, interval data, sharp identification regions, coarse data, adjunctions, partially ordered sets, linear regression model, best linear predictor, set-domained loss function

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E-mail addresses

Georg Schollmeyer		georg.schollmeyer@stat.uni-muenchen.de
Thomas Augustin		thomas.augustin@stat.uni-muenchen.de/a>