Probabilities for p-outside values – General properties

Abstract

Probability distributions are uncountably many. The task for a general and useful their classification still has no satisfactory solution. Due to lack of information outside the range of the data the tails of the distribution should be described via many characteristics. Index of regular variation is a good characteristic, but it puts too many distributions with very different tail behavior in one and the same class. One can consider for example Stable(α) and Hill-horror(α) laws with one and the same fixed parameter α > 0. When analyzing the tail behavior of the observed distribution we need some characteristic which does not depend on the moments because in the most important cases of the heavy-tailed distributions theoretical moments do not exist and the corresponding empirical moments fluctuate too much. In this paper, we show that probabilities for different types of outside values can be very appropriate characteristics of the tails of the observed distribution. They do not depend on increasing affine transformations and do not need the existence of the moments. The idea origins from Tukey’s box plots, and allows us to obtain one and the same characteristic of the tail behavior of the observed distribution within the whole distributional type with respect to all increasing affine transformations. These characteristics answer the question: “At what extent we can observe “unexpected” values?”.Probability distributions are uncountably many. The task for a general and useful their classification still has no satisfactory solution. Due to lack of information outside the range of the data the tails of the distribution should be described via many characteristics. Index of regular variation is a good characteristic, but it puts too many distributions with very different tail behavior in one and the same class. One can consider for example Stable(α) and Hill-horror(α) laws with one and the same fixed parameter α > 0. When analyzing the tail behavior of the observed distribution we need some characteristic which does not depend on the moments because in the most important cases of the heavy-tailed distributions theoretical moments do not exist and the corresponding empirical moments fluctuate too much. In this paper, we show that probabilities for different types of outside values can be very appropriate characteristics of the tails of the observed distribution. They do not depend on increasing affin...