Multi-sample means comparisons for imprecise interval data

Abstract

In recent years, interval data have become an increasingly popular tool to solving modern data problems. Intervals are now often used for dimensionality reduction, data aggregation, privacy censorship, and quantifying awareness of various uncertainties. Among many statistical methods that are being studied and developed for interval data, significance tests are of particular importance due to their fundamental value both in theory and practice. The difficulty in developing such tests mainly lies in the fact that the concept of normality does not extend naturally to intervals, making the exact tests hard to formulate. As a result, most existing works have relied on bootstrap methods to approximate null distributions. However, this is not always feasible given limited sample sizes or other intrinsic characteristics of the data. In this paper, we propose a novel asymptotic test for comparing multi-sample means with interval data as a generalization of the classic ANOVA. Based on the random sets theory, we construct the test statistic in the form of a ratio of between-group interval variance and within-group interval variance. The limiting null distribution is derived under usual assumptions and mild regularity conditions. Simulation studies with various data configurations validate the asymptotic result, and show promising small sample performances. Finally, a real interval data ANOVA analysis is presented that showcases the applicability of our method.