Symbolic Interval-Valued Regression Model

Symbolic data analysis can provide statistical inferences for macroscale data while preserving as much information as possible from microscale data. In this study, we focus on the symbolic interval-valued regression model. The microdata are reorganized into intervals by using the largest and smallest order statistics. Afterward, we develop innovative symbolic interval-valued regression models to construct the relationships between two or more intervals. Owing to the properties of order statistics, we maintain the natural order in which a higher value of the dependent variable is larger than its lower value. First, we develop a simple linear symbolic interval-valued regression model and derive the corresponding maximum likelihood estimators (MLEs). In addition, we describe the Fisher information matrix of the MLEs and show that they demonstrate asymptotic normality. Next, we extend the aforementioned model to a multiple linear symbolic interval-valued regression model, and the corresponding MLEs are again derived. Monte Carlo simulations and real data analysis confirm the validity of the proposed method.