Probabilities for asymmetric p-outside values

Since 2017, Jordanova and co-authors investigate probabilities for p-outside values and determine their explicit forms in many particular cases. They show that they are closely related to the concept of heavy tails. Tukey’s box-plots are very popular and useful in practice. In particular, the relative frequencies for observing a data point in different sections of the box-plot can help practitioners find the exact probability distribution of the underlying random variable. These open the door to working with distribution-sensitive estimators, which can be more accurate, especially in small sample investigations. All these methods, however, suffer from the drawback that they use the interquantile range in a symmetric way. This study gives greater influence on asymmetry in the analysis of the tails of distributions. New theoretical and empirical box-plots and characteristics of the tails of distributions are suggested. The theoretical asymmetric p-outside value functions do not depend on the center and scaling factor of the distribution and do not need existence of moments. Therefore, they are appropriate for comparing the tails of distributions and for estimating the parameters that govern their tail behaviour.