Multiple linear regression models for random intervals: a set arithmetic approach

Some regression models for analyzing relationships between random intervals (i.e., random variables taking intervals as outcomes) are presented. The proposed approaches are extensions of previous existing models and they account for cross relationships between midpoints and spreads (or radii) of the intervals in a unique equation based on the interval arithmetic. The estimation problem, which can be written as a constrained minimization problem, is theoretically analyzed and empirically tested. In addition, numerically stable general expressions of the estimators are provided. The main differences between the new and the existing methods are highlighted in a real-life application, where it is shown that the new model provides the most accurate results by preserving the coherency with the interval nature of the data.