Abstract
We propose a novel Bayesian analysis of the p-variate skew-t model, providing a new parameterization, a set of non-informative priors and a sampler specifically designed to explore the posterior density of the model parameters. Extensions, such as the multivariate regression model with skewed errors and the stochastic frontiers model, are easily accommodated. A novelty introduced in the paper is given by the extension of the bivariate skew-normal model given in Liseo and Parisi (2013) to a more realistic p-variate skew-t model. We also introduce the R package mvst, which produces a posterior sample for the parameters of a multivariate skew-t model.
Similar content being viewed by others
References
Azzalini A, with the collaboration of A. Capitanio (2014) The skew-normal and related families. Cambridge University Press, Cambridge
Azzalini A (2015) The R package sn: the skew-normal and skew-\(t\) distributions (version 1.3-0). Università di Padova, Padova
Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J R Stat Soc B 65:367–389
Azzalini A, Genton M (2008) Robust likelihood methods based on the skew-t and related distributions. Int Stat Rev 76:106–119
Azzalini A, Genz A (2016) The R package mnormt: the multivariate normal and \(t\) distributions (version 1.5-4). http://azzalini.stat.unipd.it/SW/Pkg-mnormt
Branco MD, Dey D (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 77:1–15
Branco MD, Genton MG, Liseo B (2011) Objective Bayesian analysis of skew-t distributions. Scand J Stat 40(1):63–85
Cabral CRB, Lachos VH, Prates MO (2012) Multivariate mixture modeling using skew-normal independent distributions. Comput Stat Data Anal 56(1):126–142
Cancho VG, Ortega EMM, Paula GA (2010) On estimation and influence diagnostics for log-Birnbaum–Saunders Student-\(t\) regression models: full Bayesian analysis. J Stat Plan Inference 140(9):2486–2496
Cappé O, Guillin A, Marin JM, Robert CP (2004) Population Monte Carlo. J Comput Graph Stat 13:907–929
Eddelbuettel D, Romain F (2013) RcppGSL: ‘Rcpp’ integration for ‘GNU GSL’ vectors and matrices. http://CRAN.R-project.org/package=RcppGSL
Fernandez C, Steel MFJ (1999) Multivariate Student-\(t\) regression models: pitfalls and inference. Biometrika 86:153–167
Fonseca TC, Ferreira MAR, Migon HS (2008) Objective Bayesian analysis for the Student-t regression model. Biometrika 95:325–333
Geisser S, Eddy WF (1979) A predictive approach to model selection. J Am Stat Assoc 74:153–160
Genton MG (2004) Skew-elliptical distributions and their applications: a Journey beyond normality. In: Genton MG (ed) CRC/Chapman & Hall/CRC, Boca Raton
Genz A, Bretz F, Miwa T, Mi X, Leisch F, Scheipl F, Hothorn T (2015) mvtnorm: multivariate normal and t distributions. R package version 1.0-3
Gough B (2009) GNU scientific library reference manual, 3rd edn. Network Theory Ltd
Gradshteyn IS, Ryzhik IM (1994) Table of integrals, series, and products. Academic Press, Inc., Boston, Russian ed. Translation edited and with a preface by Alan Jeffrey
Hansen BE (1994) Autoregressive conditional density estimation. Int Econ Rev 35(3):705–730
Ho HJ, Lin TI (2010) Robust linear mixed models using the skew \(t\) distribution with application to schizophrenia data. Biom J 52(4):449–469
Jones MC, Faddy MJ (2003) A skew extension of the t-distribution, with applications. J R Stat Soc B 65(1):159–174
Lachos VH, Ghosh P, Arellano-Valle RB (2010) Likelihood based inference for skew-normal independent linear mixed models. Stat Sin 20(1):303–322
Lee SX, McLachlan GJ (2013) EMMIXuskew: an R package for fitting mixtures of multivariate skew \(t\) distributions via the EM algorithm. J Stat Softw 55:1–22
Leisen F, Marin JM, Villa C (2016) Objective Bayesian modelling of insurance risks with the skewed Student-t distribution. arXiv:1607.04796
Li Y, Liu X, Yu J (2015) A Bayesian chi-squared test for hypothesis testing. J Econom 189(1):54–69
Liseo B, Parisi A (2013) Bayesian inference for the multivariate skew-normal model: a population Monte Carlo approach. Comput Stat Data Anal 63:125–138
Marchenko Y, Genton M (2010) A suite of commands for fitting the skew-normal and skew-t models. Stata J 10:507–539 (Cited By 2)
Martin AD, Quinn KM, Park JH (2011) MCMCpack: Markov Chain Monte Carlo in R. J Stat Softw 42:22
Prates MO, Cabral CRB, Lachos VH (2013) mixsmsn: fitting finite mixture of scale mixture of skew-normal distributions. J Stat Softw 54:1–20
R Core Team (2015) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna
Rachev ST, Hsu JSJ, Bagasheva BS, Fabozzi FJ (2008) Bayesian methods in finance. Wiley, New York
Robert CP (1995) Simulation of truncated normal variables. Stat Comput 5(2):121–125 (June 1995)
Robert CP, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer Texts in Statistics. Springer, New York
Rosco JF, Jones MC, Pewsey A (2011) Skew t distributions via the sinh-arcsinh transformation. Test 20(3):630–652
Rubio FJ, Steel MFJ (2015) Bayesian modelling of skewness and kurtosis with two-piece scale and shape distributions. Electron J Stat 9(2):1884–1912
Sartori N (2006) Bias prevention of maximum likelihood estimates for scalar skew normal and skew-t distributions. J Stat Plan Inference 136:4259–4275
StataCorp. (2015) Stata Statistical Software: Release 14
Villa C, Rubio FJ (2017) Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula. arXiv:1701.05638
Zhang JL (2014) Comparative investigation of three Bayesian p values. Comput Stat Data Anal 79:277–291
Acknowledgements
The work of B. Liseo has been supported by Sapienza Università di Roma Grant C26A15M9PC, “Rischi competitivi e organizzazione della didattica presso Sapienza: il caso degli abbandoni e dei fuori corso”. The work of A. Parisi has been supported by the project PRIN 2010–2011, Project Number 2010J3LZEN, Sector: Economics and Statistics.
Author information
Authors and Affiliations
Corresponding author
Appendices
Proof of Proposition 2.1
-
(a)
From one of the possible definitions of a multivariate ST r.v., it is known that \( U\sim \textit{SN}_p( 0, \alpha , \varOmega , \nu )\); since Y is a simple transformation of U, its distribution is readily obtained.
-
(b)
Start from \(f(y,z,v) = f(v) f(z) f(y\mid z, v)\). By assumption, f(z) is a standard Gaussian density, and
$$\begin{aligned} \left( Y \mid Z=z, V=v\right) = \left( \xi + \omega U \mid Z=z, V=v\right) = {\left\{ \begin{array}{ll} \xi + \omega X v^{-1/2} &{} z \ge 0 \\ \xi - \omega X v^{-1/2} &{} z < 0 \\ \end{array}\right. }. \end{aligned}$$Then, by using simple results on conditional Gaussian densities, one gets
$$\begin{aligned} \left( Y \mid Z=z, V=v\right) \sim \left\{ \begin{array}{ll} N_p\left( \xi + \omega \delta \displaystyle \frac{z}{\sqrt{v}}, \frac{1}{v}\omega (\varOmega - \delta \delta ^\prime ) \omega \right) &{} z \ge 0\\ N_p\left( \xi - \omega \delta \displaystyle \frac{z}{\sqrt{v}}, \frac{1}{v}\omega (\varOmega - \delta \delta ^\prime ) \omega \right) &{} z < 0\\ \end{array} \right. \end{aligned}$$Hence the result in (4).
Proposal distributions
We use the full conditional distributions as proposals for the latent variables Z and \(\xi \): each \(Z_i\) has the following full conditional distribution
where
The variables \(Z_i\) can be drawn as the product of \(Z^+_i\), a normal r.v. with parameters \(m_i\) and \(v_\theta \) truncated in 0 and the sign \(S_i\), uniform on \(\{-1, 1\}\). To generate values \(Z^+\) a rejection sampler has been employed (see Robert 1995).
The parameter \(\xi \) has the following full conditional density:
The parameters \(\psi \) and G have untractable full conditional distributions. To obtain a proposal distribution, they are approximated using only the contribution of the likelihood to the full conditional density. The parameter \(\psi \) has the following full conditional distribution
where \(\mathbb {1}_{x}(\cdot )\) denotes the indicator function. By ignoring the first two factors, we obtain the following proposal distribution
The proposal distribution has a positive density on \(\mathbb R^p\), while the full conditional is bounded on \(\varDelta _\varSigma \). This feature improves the ability of the sampler to explore the parameter space; moreover, particles which don’t respect the constraint (7) will be automatically discarded, as they have null prior (and posterior) probability density, hence a null importance weight.
The parameter G has the following full conditional density
where
Ignoring the prior term we obtain
Details about the rejection sampler
For a generic latent variable \(V_i\), the Kullback–Leibler divergence \(\textit{KL}(f||\pi _v)\) is given by
which has an analytical solution for \(\alpha _v^\star = 2C\):
This divergence has always one (and only one) minimum in \(\mathbb R^+\), given by
Rights and permissions
About this article
Cite this article
Parisi, A., Liseo, B. Objective Bayesian analysis for the multivariate skew-t model. Stat Methods Appl 27, 277–295 (2018). https://doi.org/10.1007/s10260-017-0404-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-017-0404-0