Generalized weibull family: a structural analysis

Abstract

A two shape parameter generalization of the well known family of the Weibull distributions is presented and its properties are studied. The properties examined include the skewness and kurtosis, density shapes and tail character, and relation of the members of the family to those of the Pear-sonian system. The members of the family are grouped in four classes in terms of these properties. Also studied are the extreme value distributions and the limiting distributions of the extreme spacings for the members of the family. It is seen that the generalized Weibull family contains distributions with a variety of density and tail shapes, and distributions which in terms of skewness and kurtosis approximate the main types of curves of the Pearson system. Furthermore, as shown by the extreme value and extreme spacings distributions the family contains short, medium and long tailed distributions. The quantile and density quantile functions are the principle tools used for the structural analysis of the family.

Keywords:

• density quantile function
• extreme value distributions kurtosis
• pearson system
• quantile function
• skewness