On the consistency of a spatial-type interval-valued median for random intervals

TL;DR

This paper proves that the sample $d_\theta$-median is a strongly consistent estimator of the true median for interval-valued random variables, offering a robust alternative to the mean.

Contribution

It establishes the strong consistency of the spatial-type $d_\theta$-median estimator under general conditions for interval-valued data.

Findings

The $d_\theta$-median is robust against outliers.

The estimator is strongly consistent.

Applicable under broad conditions.

Abstract

The sample $d_\theta$-median is a robust estimator of the central tendency or location of an interval-valued random variable. While the interval-valued sample mean can be highly influenced by outliers, this spatial-type interval-valued median remains much more reliable. In this paper, we show that under general conditions the sample $d_\theta$-median is a strongly consistent estimator of the $d_\theta$-median of an interval-valued random variable.