Web-based Supplementary Materials for “Bayesian Semiparametric Nonlin- ear Mixed-Effects Joint Models for Data with Skewness, Missing Responses and Measurement Errors in Covariates” by Y. Huang and G. Dagne Web Appendix A. Multivariate Skew Distributions

Web Appendix A. Multivariate Skew Distributions Different versions of the multivariate skew–normal(SN) and skew-t (ST) distributions have been considered and used in the literature (Arellano-Valle and Genton, 2005; Arellano-Valle et al, 2007; Azzalini and Capitanio, 2003; Sahu et al, 2003 and among others). A new class of distributions by introducing skewness in multivariate elliptically distributions were developed in publication (Sahu et al, 2003). The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate SN and ST distributions. For completeness, this appendix briefly summarizes the multivariate SN and ST distributions that will be used in defining the skew BNLME joint models considered in this paper. Assume an m-dimensional random vector Y follows an m variate SN or ST distribution with location vector μ, m ×m positive (diagonal) dispersion matrix Σ and m ×m skewness matrix ∆(δ) = diag(δ1, δ2, . . . , δm) or the degrees of freedom ν, where δ = (δ1, . . . , δm) T . In what follows, we briefly discuss multivariate SN and ST distributions introduced by Sahu et al(2003) which are suitable for a Bayesian inference since they is built using conditional method. For detailed discussions on properties of SN and ST distributions, see Reference (Sahu et al, 2003).