A mixture of generalized hyperbolic distributions

Abstract

We introduce a mixture of generalized hyperbolic distributions as an alternative to the ubiquitous mixture of Gaussian distributions as well as their near relatives of which the mixture of multivariate t and skew-t distributions are predominant. The mathematical development of our mixture of generalized hyperbolic distributions model relies on its relationship with the generalized inverse Gaussian distribution. The latter is reviewed before our mixture models are presented along with details of the aforesaid reliance. Parameter estimation is outlined within the expectation-maximization framework before the clustering performance of our mixture models is illustrated via applications on simulated and real data. In particular, the ability of our models to recover parameters for data from underlying Gaussian and skew-t distributions is demonstrated. Finally, the role of Generalized hyperbolic mixtures within the wider modelbased clustering, classification, and density estimation literature is discussed.

Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering

“…Also we observed (supplementary Fig. 11) that the GH parameterization proposed in Browne and McNicholas (2013) provided results very close to the standard NIG distribution (Figs. 2 (a) and 3 (c)).…”

Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers

“…Instead of applying transformations, we could extend our proposal by considering conditional skew distributions for each state, and this should be in line with the growing interest in proposing mixture models where the component distributions are skewed. Examples of existing approaches in this direction are: mixtures of skew-normal distributions (Lin, 2009 andPyne et al, 2009), mixtures of shifted asymmetric Laplace distributions (Franczak et al, 2014), mixtures of multivariate skew-t distributions (see, e.g., Lin, 2010, andMcLachlan, 2014), mixtures of multivariate t distributions with the Box-Cox transformation (Lo and Gottardo, 2012), mixtures of multivariate normal inverse Gaussian distributions (Karlis and Santourian, 2009), and mixtures of generalized hyperbolic distributions (Browne and McNicholas, 2015). For a recent enough survey about non-elliptical distributions in mixture modelling, see Lee and McLachlan (2013).…”

A bootstrap‐augmented alternating expectation‐conditional maximization algorithm for mixtures of factor analyzers

“…Mixture models provide a statistically sound option for the task of unsupervised learning (McNicholas, ). Although models with increased flexibility through alternative choices of probability densities are increasingly available (Andrews, Wickins, Boers, & McNicholas, ; Browne & McNicholas, ; Franczak, Browne, & McNicholas, ; Vrbik & McNicholas, ), the bulk of attention remains on mixtures of multivariate Gaussian distributions (Banfield & Raftery, ; Celeux & Govaert, ; Fop, Murphy, & Scrucca, ; Fraley & Raftery, ; McNicholas & Murphy, ; Scrucca, Fop, Murphy, & Raftery, ). This manuscript continues in the latter vein, assuming that the distribution of the random vector X takes the form where ϕ denotes the multivariate Gaussian density function and 𝛍 g and Σ g represent the mean and covariance matrix of the respective group g .…”