On a Notion of Outliers Based on Ratios of Order Statistics

Abstract

There are a number of mathematical formalisms of the term "outlier" in statistics, though there is no consensus on what the right notion ought to be. Accordingly, we try to give a consistent and robust definition for a specific type of outliers defined via order statistics. Our approach is based on ratios of partial sums of order statistics to investigate the tail behaviors of hypothetical and empirical distributions. We simulate our statistic on a set of distributions to mark potential outliers and use an algorithm to automatically select a cut-off point without the need of any further a priori assumption. Finally, we show the efficacy of our statistic by a simulation study on distinguishing two Pareto tails outside of the L\'{e}vy stable region.