Comparing the Medians of a Random Interval Defined by Means of Two Different L ¹ Metrics

Abstract

The standard central tendency measure for interval-valued data is the Aumann-type expected value, but as in real settings it is not always convenient because of the big influence that small changes in the data as well as the existence of great magnitude data have on its estimate. The aim of this paper is to explore other summary measures with a more robust behavior. The real-valued case has served as inspiration to define the median of a random interval. The definition of the median as a ‘middle position’ value is not possible here because of the lack of a universally accepted total order in the space of interval data, so the median is defined as the element which minimizes the mean distance, in terms of an L ¹ metric (extension of the Euclidean distance in ℝ), to the values the random interval can take. The two metrics that we consider are the generalized Hausdorff metric (like the well-known Hausdorff metric, but including a positive parameter which determines the relative importance given to the difference in imprecision with respect to the difference in location) and the 1-norm metric introduced by Vitale. The aim of this paper is to compare these two approaches for the median of a random interval, both theoretically based on concepts commonly used in robustness and empirically by simulation.