Generalized Univariate Distributions and a New Asymmetric Laplace Model

This work provides a survey of general class of distributions generated from a mixture of beta random variables. We provide an extensive review of the literature, concerning generating new distributions via the inverse CDF transformation. In particular, we account for beta generated and Kumaraswamy generated families of distributions. We provide a brief summary of each of their families of distributions. We also propose a new asymmetric mixture distribution, which is an alternative to beta generated distributions. We provide basic properties of this new class of distributions generated from the Laplace model. We also address the issue of parameter estimation of this new skew generalized Laplace model

Rethinking Generalized Beta Family of Distributions

We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the "traditional" GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the "traditional" GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S\&P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings.Comment: 21 pages, 11 figyes, 2 table

Some Extended Classes of Distributions: Characterizations and Properties

Based on a simple relationship between two truncated moments and certain functions of the th order statistic, we characterize some extended classes of distributions recently proposed in the statistical literature, videlicet Beta-G, Gamma-G, Kumaraswamy-G and McDonald-G. Several properties of these extended classes and some special cases are discussed. We compare these classes in terms of goodness-of-fit criteria using some baseline distributions by means of two real data sets