Bayesian modelling of skewness and kurtosis with two-piece scale and shape distributions

We formalise and generalise the definition of the family of univariate double two--piece distributions, obtained by using a density--based transformation of unimodal symmetric continuous distributions with a shape parameter. The resulting distributions contain five interpretable parameters that control the mode, as well as the scale and shape in each direction. Four-parameter subfamilies of this class of distributions that capture different types of asymmetry are discussed. We propose interpretable scale and location-invariant benchmark priors and derive conditions for the propriety of the corresponding posterior distribution. The prior structures used allow for meaningful comparisons through Bayes factors within flexible families of distributions. These distributions are applied to data from finance, internet traffic and medicine, comparing them with appropriate competitors

Flexible modelling in statistics: past, present and future

In times where more and more data become available and where the data exhibit rather complex structures (significant departure from symmetry, heavy or light tails), flexible modelling has become an essential task for statisticians as well as researchers and practitioners from domains such as economics, finance or environmental sciences. This is reflected by the wealth of existing proposals for flexible distributions; well-known examples are Azzalini's skew-normal, Tukey's $g$-and-$h$, mixture and two-piece distributions, to cite but these. My aim in the present paper is to provide an introduction to this research field, intended to be useful both for novices and professionals of the domain. After a description of the research stream itself, I will narrate the gripping history of flexible modelling, starring emblematic heroes from the past such as Edgeworth and Pearson, then depict three of the most used flexible families of distributions, and finally provide an outlook on future flexible modelling research by posing challenging open questions.Comment: 27 pages, 4 figure

Natural (non-)informative priors for skew-symmetric distributions

In this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew-symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the Total Variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our noninformative priors induce posterior distributions with good frequentist properties, similar to those of the Jeffreys prior. Our informative priors yield better results than their competitors from the literature. We also propose a scale- and location-invariant prior structure for models with unknown location and scale parameters and provide sufficient conditions for the propriety of the corresponding posterior distribution. Illustrative examples are presented using simulated and real data.Comment: 30 pages, 3 figure