Non-Gaussian Multivariate Statistical Models and their Applications

In recent years, the extension of classical Gaussian-based statistical models to non-Gaussian ones has received sustained attention and generated many research topics. There have been various approaches to develop multivariate non-Gaussian distributions. Two main directions to approach such extensions consist of multivariate skew-symmetric distributions and of copula models. Skew-symmetric distributions are extensions of the multivariate Gaussian distribution to model both skewness and heavy tails; see the book edited by Genton (2004) and the review article by Azzalini (2005) for more details. They include the multivariate normal distribution as a particular case and have many appealing properties. Moreover they are flexible for modeling of data from real applications and can be used in practical settings. Indeed, those skew-symmetric multivariate models have been applied to problems such as: non-Gaussian Kalman filters to analyze climatic time series data; hierarchical Bayesian spatial models for rainfall data; mixed models with non-Gaussian random effects to study biomedical data; clinical trials where one of several doses or treatments is selected in a phase II study to be examined further in a phase III study and naturally give rise to skew-symmetric models; non-Gaussian asset pricing and portfolio selection in finance; non-Gaussian GARCH models in economics; non-Gaussian shape modeling in image analysis; nonGaussian modeling of coastal flooding in oceanography; non-Gaussian distance determination in astronomy; non-Gaussian space time regression modeling of tree growth in forestry; sample selection bias in Heckmantype selection models in economics; perturbation of numerical confidential non-Gaussian data for database security in management science; non-Gaussian modeling of the distribution of pollutants in environmetrics; models for non-Gaussian distributions of wind speed and direction in wind power forecasting for renewable energy; and many more. In the past six years, many new results have been obtained and their applications have been expanding. Somewhat in parallel, copula models have been developed to describe the joint dependence structure of multivariate distributions separately from their marginal distributions; see Joe (1997), McNeil et al. (2005), Nelsen (2006), Jaworski et al. (2010), Kurowicka and Cooke (2006), Kurowicka and Joe (2011) for overviews. Also there are other books aimed at readers in quantitative finance and insurance that have chapters on copulas. In applications in finance and insurance, models with more dependence in the joint tails than multivariate non-Gaussian are important. This explains why copula models which can allow for tail dependence have become especially important in finance and insurance. Modeling multivariate data with copulas has found many