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

Statistical modelling under epistemic data imprecision : some results on estimating multinomial distributions and logistic regression for coarse categorical data

Paper presented at 9th International Symposium on Imprecise Probability: Theories and Applications, Pescara, Italy, 2015. Abstract: The paper deals with parameter estimation for categorical data under epistemic data imprecision, where for a part of the data only coarse(ned) versions of the true values are observable. For different observation models formalizing the information available on the coarsening process, we derive the (typically set-valued) maximum likelihood estimators of the underlying distributions. We discuss the homogeneous case of independent and identically distributed variables as well as logistic regression under a categorical covariate. We start with the imprecise point estimator under an observation model describing the coarsening process without any further assumptions. Then we determine several sensitivity parameters that allow the refinement of the estimators in the presence of auxiliary information

Detecting stochastic dominance for poset-valued random variables as an example of linear programming on closure systems

In this paper we develop a linear programming method for detecting stochastic dominance for random variables with values in a partially ordered set (poset) based on the upset-characterization of stochastic dominance. The proposed detection-procedure is based on a descriptively interpretable statistic, namely the maximal probability-difference of an upset. We show how our method is related to the general task of maximizing a linear function on a closure system. Since closure systems are describable via their valid formal implications, we can use here ingredients of formal concept analysis. We also address the question of inference via resampling and via conservative bounds given by the application of Vapnik-Chervonenkis theory, which also allows for an adequate pruning of the envisaged closure system that allows for the regularization of the test statistic (by paying a price of less conceptual rigor). We illustrate the developed methods by applying them to a variety of data examples, concretely to multivariate inequality analysis, item impact and differential item functioning in item response theory and to the analysis of distributional differences in spatial statistics. The power of regularization is illustrated with a data example in the context of cognitive diagnosis models

Detecting stochastic dominance for poset-valued random variables as an example of linear programming on closure systems

In this paper we develop a linear programming method for detecting stochastic dominance for random variables with values in a partially ordered set (poset) based on the upset-characterization of stochastic dominance. The proposed detection-procedure is based on a descriptively interpretable statistic, namely the maximal probability-difference of an upset. We show how our method is related to the general task of maximizing a linear function on a closure system. Since closure systems are describable via their valid formal implications, we can use here ingredients of formal concept analysis. We also address the question of inference via resampling and via conservative bounds given by the application of Vapnik-Chervonenkis theory, which also allows for an adequate pruning of the envisaged closure system that allows for the regularization of the test statistic (by paying a price of less conceptual rigor). We illustrate the developed methods by applying them to a variety of data examples, concretely to multivariate inequality analysis, item impact and differential item functioning in item response theory and to the analysis of distributional differences in spatial statistics. The power of regularization is illustrated with a data example in the context of cognitive diagnosis models

Concepts for Decision Making under Severe Uncertainty with Partial Ordinal and Partial Cardinal Preferences

We introduce three different approaches for decision making under uncertainty if (I) there is only partial (both cardinally and ordinally scaled) information on an agent’s preferences and (II) the uncertainty about the states of nature is described by a credal set (or some other imprecise probabilistic model). Particularly, situation (I) is modeled by a pair of binary relations, one specifying the partial rank order of the alternatives and the other modeling partial information on the strength of preference. Our first approach relies on decision criteria constructing complete rankings of the available acts that are based on generalized expectation intervals. Subsequently, we introduce different concepts of global admissibility that construct partial orders between the available acts by comparing them all simultaneously. Finally, we define criteria induced by suitable binary relations on the set of acts and, therefore, can be understood as concepts of local admissibility. For certain criteria, we provide linear programming based algorithms for checking optimality/admissibility of acts. Additionally, the paper includes a discussion of a prototypical situation by means of a toy example

A Probabilistic Evaluation Framework for Preference Aggregation Reflecting Group Homogeneity

Groups differ in the homogeneity of their members' preferences. Reflecting this, we propose a probabilistic criterion for evaluating and comparing the adequateness of preference aggregation procedures that takes into account information on the considered group's homogeneity structure. Further, we discuss two approaches for approximating our criterion if information is only imperfectly given and show how to estimate these approximations from data. As a preparation, we elaborate some general minimal requirements for measuring homogeneity and discuss a specific proposal for a homogeneity measure. Finally, we investigate our framework by comparing aggregation rules in a simulation study

A simple descriptive method for multidimensional item response theory based on stochastic dominance

In this paper we develop a descriptive concept of a (partially) ordinal joint scaling of items and persons in the context of (dichotomous) item response analysis. The developed method has to be understood as a purely descriptive method describing relations among the data observed in a given item response data set, it is not intended to directly measure some presumed underlying latent traits. We establish a hierarchy of pairs of item difficulty and person ability orderings that empirically support each other. The ordering principles we use for the construction are essentially related to the concept of first order stochastic dominance. Our method is able to avoid a paradoxical result of multidimensional item response theory models described in \cite{hooker2009paradoxical}. We introduce our concepts in the language of formal concept analysis. This is due to the fact that our method has some similarities with formal concept analysis and knowledge space theory: Both our methods as well as descriptive techniques used in knowledge space theory (concretely, item tree analysis) could be seen as two different stochastic generalizations of formal implications from formal concept analysis

A simple descriptive method for multidimensional item response theory based on stochastic dominance

In this paper we develop a descriptive concept of a (partially) ordinal joint scaling of items and persons in the context of (dichotomous) item response analysis. The developed method has to be understood as a purely descriptive method describing relations among the data observed in a given item response data set, it is not intended to directly measure some presumed underlying latent traits. We establish a hierarchy of pairs of item difficulty and person ability orderings that empirically support each other. The ordering principles we use for the construction are essentially related to the concept of first order stochastic dominance. Our method is able to avoid a paradoxical result of multidimensional item response theory models described in \cite{hooker2009paradoxical}. We introduce our concepts in the language of formal concept analysis. This is due to the fact that our method has some similarities with formal concept analysis and knowledge space theory: Both our methods as well as descriptive techniques used in knowledge space theory (concretely, item tree analysis) could be seen as two different stochastic generalizations of formal implications from formal concept analysis

Quantifying Degrees of E-admissibility in Decision Making with Imprecise Probabilities

This paper is concerned with decision making using imprecise probabilities. In the first part, we introduce a new decision criterion that allows for explicitly modeling how far decisions that are optimal in terms of Walley’s maximality are accepted to deviate from being optimal in the sense of Levi’s E-admissibility. For this criterion, we also provide an efficient and simple algorithm based on linear programming theory. In the second part of the paper, we propose two new measures for quantifying the extent of E-admissibility of an E-admissible act, i.e. the size of the set of measures for which the corresponding act maximizes expected utility. The first measure is the maximal diameter of this set, while the second one relates to the maximal barycentric cube that can be inscribed into it. Also here, for both measures, we give linear programming algorithms capable to deal with them. Finally, we discuss some ideas in the context of ordinal decision theory. The paper concludes with a stylized application examples illustrating all introduced concepts

A Probabilistic Evaluation Framework for Preference Aggregation Reflecting Group Homogeneity

Groups differ in the homogeneity of their members' preferences. Reflecting this, we propose a probabilistic criterion for evaluating and comparing the adequateness of preference aggregation procedures that takes into account information on the considered group's homogeneity structure. Further, we discuss two approaches for approximating our criterion if information is only imperfectly given and show how to estimate these approximations from data. As a preparation, we elaborate some general minimal requirements for measuring homogeneity and discuss a specific proposal for a homogeneity measure. Finally, we investigate our framework by comparing aggregation rules in a simulation study