{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T02:31:57Z","timestamp":1783996317087,"version":"3.55.0"},"reference-count":59,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,3,25]],"date-time":"2020-03-25T00:00:00Z","timestamp":1585094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Bijzonder Onderzoeksfonds","award":["N\/A"],"award-info":[{"award-number":["N\/A"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Skew-symmetric distributions are a popular family of flexible distributions that conveniently model non-normal features such as skewness, kurtosis and multimodality. Unfortunately, their frequentist inference poses several difficulties, which may be adequately addressed by means of a Bayesian approach. This paper reviews the main prior distributions proposed for the parameters of skew-symmetric distributions, with special emphasis on the skew-normal and the skew-t distributions which are the most prominent skew-symmetric models. The paper focuses on the univariate case in the absence of covariates, but more general models are also discussed.<\/jats:p>","DOI":"10.3390\/sym12040491","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"491","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Bayesian Inference for Skew-Symmetric Distributions"],"prefix":"10.3390","volume":"12","author":[{"given":"Fatemeh","family":"Ghaderinezhad","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281, S9, Campus Sterre, 9000 Gent, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2290-8437","authenticated-orcid":false,"given":"Christophe","family":"Ley","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281, S9, Campus Sterre, 9000 Gent, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicola","family":"Loperfido","sequence":"additional","affiliation":[{"name":"Dipartimento di Economia, Societ\u00e0 e Politica, Universit\u00e0 degli Studi di Urbino \u201cCarlo Bo\u201d, Via Saffi 42, 61029 Urbino (PU), Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,25]]},"reference":[{"key":"ref_1","unstructured":"El-Shaarawi, A., and Piegorsch, W. (2012). Skew distributions. Statistical Theory and Methods, Encyclopedia of Environmetrics, Wiley. [2nd ed.]."},{"key":"ref_2","first-page":"171","article-title":"A class of distributions which includes the normal ones","volume":"12","author":"Azzalini","year":"1985","journal-title":"Scand. J. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Azzalini, A., and Capitanio, A. (2014). The Skew-Normal and Related Families, Cambridge University Press.","DOI":"10.1017\/CBO9781139248891"},{"key":"ref_4","first-page":"1259","article-title":"A skew-symmetric representation of multivariate distribution","volume":"14","author":"Wang","year":"2004","journal-title":"Stat. Sin."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1111\/1467-9868.00391","article-title":"Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution","volume":"65","author":"Azzalini","year":"2003","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_6","first-page":"199","article-title":"Further results on a class of distributions which includes the normal ones","volume":"46","author":"Azzalini","year":"1986","journal-title":"Statistica"},{"key":"ref_7","first-page":"197","article-title":"The skew logistic distribution","volume":"93","author":"Nadarajah","year":"2009","journal-title":"Statistica"},{"key":"ref_8","first-page":"271","article-title":"A probabilistic representation of the skew-normal distribution","volume":"13","author":"Henze","year":"1986","journal-title":"Scand. J. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1111\/1467-9868.00194","article-title":"Statistical applications of the multivariate skew-normal distribution","volume":"61","author":"Azzalini","year":"1999","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1111\/j.1751-5823.2007.00016.x","article-title":"Robust likelihood methods based on the skew-t and related distributions","volume":"76","author":"Azzalini","year":"2008","journal-title":"Int. Stat. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s10260-005-0117-7","article-title":"A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution","volume":"14","author":"Chiogna","year":"2005","journal-title":"Stat. Methods Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"747","DOI":"10.3150\/12-BEJ346","article-title":"Skew-symmetric distributions and Fisher information\u2014A tale of two densities","volume":"18","author":"Hallin","year":"2012","journal-title":"Bernoulli"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1432","DOI":"10.3150\/13-BEJ528","article-title":"Skew-symmetric distributions and Fisher information: The double sin of the skew-normal","volume":"20","author":"Hallin","year":"2014","journal-title":"Bernoulli"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1016\/j.jmva.2009.10.008","article-title":"On the singularity of multivariate skew-symmetric models","volume":"101","author":"Ley","year":"2010","journal-title":"J. Multivar. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1080\/02664760050120542","article-title":"Problems of inference for Azzalini\u2019s skew-normal distribution","volume":"27","author":"Pewsey","year":"2000","journal-title":"J. Appl. Stat."},{"key":"ref_16","first-page":"1362","article-title":"The centred parametrization for the multivariate skew-normal distribution","volume":"99","author":"Azzalini","year":"2009","journal-title":"J. Multivar. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/biomet\/80.1.27","article-title":"Bias reduction of maximum likelihood estimates","volume":"80","author":"Firth","year":"1993","journal-title":"Biometrika"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2013). Bayesian Data Analysis, Chapman & Hall\/CRC. [3rd ed.].","DOI":"10.1201\/b16018"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1111\/j.2517-6161.1979.tb01066.x","article-title":"Reference posterior distributions for Bayesian inference (with discussion)","volume":"41","author":"Bernardo","year":"1979","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_20","first-page":"370","article-title":"An essay towards solving a problem in the doctrine of chances","volume":"53","author":"Bayes","year":"1763","journal-title":"Philos. Trans. R. Soc. Ser. A"},{"key":"ref_21","unstructured":"Laplace, P.S. (1812). Th\u00e9orie Analytique des Probabilit\u00e9s, Courcier."},{"key":"ref_22","first-page":"35","article-title":"On the development of reference priors (with discussion)","volume":"4","author":"Bernardo","year":"1992","journal-title":"Bayesian Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00031305.1986.10475342","article-title":"Why isn\u2019t everyone a Bayesian? (with discussion)","volume":"40","author":"Efron","year":"1986","journal-title":"Am. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"986","DOI":"10.1214\/aoms\/1177728069","article-title":"On the measure of the information provided by an experiment","volume":"27","author":"Lindley","year":"1956","journal-title":"Ann. Math. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1080\/01621459.1989.10478756","article-title":"Estimating a product of means. Bayesian analysis with reference priors","volume":"84","author":"Berger","year":"1989","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1111\/j.2517-6161.1958.tb00278.x","article-title":"Fiducial distributions and Bayes\u2019 theorem","volume":"20","author":"Lindley","year":"1958","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Datta, G.S., and Mukerjee, R. (2004). Probability Matching Priors: Higher Order Asymptotics, Springer.","DOI":"10.1007\/978-1-4612-2036-7"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1111\/j.2517-6161.1963.tb00512.x","article-title":"On formulae for confidence points based on integrals of weighted likelihoods","volume":"25","author":"Welch","year":"1963","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_29","first-page":"187","article-title":"Objective priors: an introduction for frequentists","volume":"26","author":"Ghosh","year":"2011","journal-title":"J. Stat. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1093\/biomet\/80.3.499","article-title":"Frequentist validity of posterior quantiles in the presence of a nuisance parameter: Higher order asymptotics","volume":"80","author":"Mukerjee","year":"1993","journal-title":"Biometrika"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1093\/biomet\/84.4.970","article-title":"Second-order probability matching priors","volume":"84","author":"Mukerjee","year":"1997","journal-title":"Biometrika"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.jspi.2004.06.062","article-title":"A note on reference priors for the scalar skew-normal distribution","volume":"136","author":"Liseo","year":"2006","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1111\/j.1467-9469.2011.0775.x","article-title":"A matching prior for the shape parameter of the skew-normal distribution","volume":"39","author":"Cabras","year":"2012","journal-title":"Scand. J. Stat."},{"key":"ref_34","unstructured":"Canale, A., and Scarpa, B. (2013). Informative Bayesian inference for the skew-normal distribution. arXiv."},{"key":"ref_35","first-page":"141","article-title":"Bayesian inference for the skewness parameter of the scalar skew-normal distribution","volume":"21","author":"Bayes","year":"2007","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_36","unstructured":"Chaibub Neto, E., and Branco, M.D. (2003). Bayesian Reference Analysis for Binomial Calibration Problem, IME-USP. Technical Report."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1093\/biomet\/70.2.343","article-title":"On a formula for the distribution of the maximum likelihood estimator","volume":"70","year":"1983","journal-title":"Biometrika"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3539","DOI":"10.1016\/j.jspi.2005.01.014","article-title":"Likelihood based discrimination between separate regression models","volume":"136","author":"Pace","year":"2006","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1198\/jasa.2009.0133","article-title":"Prior distributions from pseudo-likelihoods in the presence of nuisance parameters","volume":"104","author":"Ventura","year":"2009","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1111\/j.1467-9469.2006.00503.x","article-title":"On the unification of families of skew-normal distributions","volume":"33","author":"Azzalini","year":"2006","journal-title":"Scand. J. Stat."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3015","DOI":"10.1080\/02664763.2016.1157144","article-title":"Bayesian modelling of university first-year students\u2019 grades after placement test","volume":"43","author":"Canale","year":"2016","journal-title":"J. Appl. Stat."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1198\/016214501750332848","article-title":"Marginal likelihood from the Metropolis-Hastings output","volume":"96","author":"Chib","year":"2001","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_43","unstructured":"Colubi, A. (2012, January 27\u201331). Bayesian analysis of a skewed exponential power distribution. Proceedings of the Computational Statistics 2012, Limassol, Cyprus."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"696","DOI":"10.22237\/jmasm\/1478003880","article-title":"Bayesian analysis of discrete skewed Laplace distribution","volume":"15","author":"Hossianzadeh","year":"2016","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1093\/biomet\/86.1.153","article-title":"Multivariate Student t regression models: pitfalls and inference","volume":"86","author":"Steel","year":"1999","journal-title":"Biometrika"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4259","DOI":"10.1016\/j.jspi.2005.08.043","article-title":"Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions","volume":"136","author":"Sartori","year":"2008","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1111\/j.1467-9469.2011.00779.x","article-title":"Objective Bayesian analysis of skew-t distributions","volume":"40","author":"Branco","year":"2012","journal-title":"Scand. J. Stat."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.spl.2013.11.012","article-title":"On the independence Jeffreys prior for skew-symmetric models","volume":"85","author":"Rubio","year":"2014","journal-title":"Stat. Probab. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1111\/sjos.12306","article-title":"Natural (non-)informative priors for skew-symmetric distributions","volume":"45","author":"Dette","year":"2018","journal-title":"Scand. J. Stat."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1214\/15-EJS1060","article-title":"Bayesian modelling of skewness and kurtosis with two-piece scale and shape distributions","volume":"9","author":"Rubio","year":"2015","journal-title":"Electron. J. Stat."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2441","DOI":"10.1002\/sim.6897","article-title":"Bayesian linear regression with skew-symmetric error distributions with applications to survival analysis","volume":"35","author":"Rubio","year":"2016","journal-title":"Stat. Med."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.csda.2013.02.007","article-title":"Bayesian inference for the multivariate skew-normal model: A population Monte Carlo approach","volume":"63","author":"Liseo","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s10260-017-0404-0","article-title":"Objective Bayesian analysis for the multivariate skew-t model","volume":"27","author":"Parisi","year":"2017","journal-title":"Stat. Methods Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1016\/j.ijforecast.2008.08.009","article-title":"Bayesian density forecasting of intraday electricity prices using multivariate skew t distributions","volume":"24","author":"Panagiotelisa","year":"2008","journal-title":"Int. J. Forecast."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1093\/biostatistics\/kxp062","article-title":"Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions","volume":"11","author":"Pyne","year":"2010","journal-title":"Biostatistics"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"129","DOI":"10.2307\/3316064","article-title":"A new class of multivariate skew distributions with applications to Bayesian regression models","volume":"31","author":"Sahu","year":"2003","journal-title":"Can. J. Stat."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1080\/02664760701236905","article-title":"Bayesian inference for skew-normal linear mixed models","volume":"34","author":"Bolfarine","year":"2007","journal-title":"J. Appl. Stat."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1214\/18-STS666","article-title":"Recent progress in log-concave density estimation","volume":"33","author":"Samworth","year":"2018","journal-title":"Stat. Sci."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02589075","article-title":"Block unimodality for multivariate Bayesian robustness","volume":"2","author":"Liseo","year":"1993","journal-title":"J. Ital. Stat. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/491\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:11:23Z","timestamp":1760173883000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/491"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,25]]},"references-count":59,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040491"],"URL":"https:\/\/doi.org\/10.3390\/sym12040491","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,25]]}}}