Tailweight with respect to the mode for unimodal distributions
Location, scale, skewness and tailweight are important concepts for the description of a probability distribution. The study of tails for skewed distributions presents a particular aspect for unimodal distributions: when we consider the graph of the probability density function, the mode seems to be an appealing centre. An analogous approach to the quantile-based description of \(F\) is envisaged; it takes the mode as reference and makes use of \(A_F\) and \(Q_F\) as location and dispersion functional parameters. The mode, as location parameter, does not preserve the stochastic ordering. However, its interpretation makes it more rational in certain circumstances than mean or median. So, a description with respect to the mode of a distribution would be welcomed. Basic definitions and notations are given in Section 2. Section 3 then deals with the concept of location with respect to the mode by making use of the modal central interval and its midpoint. A way to approach distributions in the mode-sense, using already known notions in the median-sense, is suggested. Section 4 concerns tailweight orderings and skewness. A location- and scale-free tailweight ordering is introduced to compare tail with respect to the mode, without assuming symmetry. Finally, mode-based concepts of qualitative aspects of skewness are considered.
scientific article; zbMATH DE number 926965
scientific article