José T.A.S. Ferreira

A Constructive Representation of Univariate Skewed Distributions

We introduce a general perspective on the introduction of skewness into symmetric distributions. Through inverse probability integral transformations we provide a constructive representation of skewed distributions, where the skewing mechanism and the original symmetric distributions are specified separately. We study the effects of the skewing mechanism on, e.g., modality, tail behavior and the amount of skewness generated. The representation is used to introduce novel classes of skewed distributions, where we induce certain prespecified characteristics through particular choices of the skewing mechanism. Finally, we use a Bayesian linear regression framework to compare the new classes with some existing distributions in the context of two empirical examples.

Directional log-spline distributions

We introduce a new class of distributions to model directional data, based on hyperspherical log-splines. The class is very flexible and can be used to model data that exhibits features that cannot be accommodated by typical parametric distributions, such as asymmetries and multimodality. The distributions are defined on hyperspheres of any dimension and thus, include the most common circular and spherical cases. Due to the flexibility of hyperspherical log-splines, the distributions can approximate well the distribution of any phenomenon and are as smooth as desired. We propose a Bayesian setup for conducting inference with directional log-spline distributions where we pay particular attention to the prior specification and the matching of the priors of the log-splines model and the model constructed through a mixture of von Mises distributions. We compare both models in the context of three data sets: generated data on the circle, a circular application concerning the movement of turtles and a spherical application on the arrival direction of cosmic rays.

Flexible univariate continuous distributions

Based on a constructive representation, which distinguishes between a skewing mechanism P and an underlying symmetric distribution F, we introduce two flexible classes of distributions. They are generated by nonparametric modelling of either P or F. We examine properties of these distributions and consider how they can help us to identify which aspects of the data are badly captured by simple symmetric distributions. Within a Bayesian framework, we investigate useful prior settings and conduct inference through MCMC methods. On the basis of simulated and real data examples, we make recommendations for the use of our models in practice. Our models perform well in the context of density estimation using the multimodal galaxy data and for regression modelling with data on the body mass index of athletes.