Linear Regression for Interval-Valued Data: A New and Comprehensive Model

• Activity Number: 26
• Type: Contributed
• Date/Time: Sunday, August 3, 2014 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Yan Sun and Chunyang Li*+
• Companies: Utah State University and
• Keywords: interval linear regression ; random interval ; least squares estimate ; asymptotic unbiasedness ; analysis of errors ; metric space

Abstract

We introduce a new probabilistic linear regression model for interval-valued data that integrates and improves over the previous results. Based on the random set theory, our model captures the geometric structure of the random intervals in a natural and rigorous way. It also connects to the classical linear regression theory by sharing several important properties. Furthermore, it accommodates an analysis of errors that provides a thorough understanding of the randomness in the interval-valued data. To estimate the model parameters, we carry out theoretical investigations of the least squares (LS) method that is widely used in the literature of interval-valued statistics. Particularly, we find the explicit LS solution that exists with probability going to one. In addition, the theoretical properties we obtain regarding the LS method are very useful for future development of more refined estimators, especially for small sample sizes. Our simulation shows that performances of the LS estimators are good for moderate sample sizes. An application to a climate data set is provided to demonstrate the applicability of our model and method.