Rubio, Francisco Javier and Steel, Mark F. J. (2014): Bayesian modelling of skewness and kurtosis with twopiece scale and shape transformations.

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Abstract
We introduce the family of univariate double two–piece distributions, obtained by using a density– based transformation of unimodal symmetric continuous distributions with a shape parameter. The resulting distributions contain five interpretable parameters that control the mode, as well as the scale and shape in each direction. Fourparameter subfamilies of this class of distributions that capture different types of asymmetry are presented. We propose interpretable scale and locationinvariant benchmark priors and derive conditions for the existence of the corresponding posterior distribution. The prior structures used allow for meaningful comparisons through Bayes factors within flexible families of distributions. These distributions are applied to models in finance, internet traffic data, and medicine, comparing them with appropriate competitors.
Item Type:  MPRA Paper 

Original Title:  Bayesian modelling of skewness and kurtosis with twopiece scale and shape transformations 
Language:  English 
Keywords:  model comparison; posterior existence; prior elicitation; scale mixtures of normals; unimodal continuous distributions 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C11  Bayesian Analysis: General ?? C16 ?? 
Item ID:  57102 
Depositing User:  Mark F.J. Steel 
Date Deposited:  05 Jul 2014 06:14 
Last Modified:  27 Sep 2019 06:39 
References:  Aas, K., and Haff, I. H. (2006), “The Generalized Hyperbolic Skew Student’s t−distribution,” Journal of Financial Econometrics, 4, 275–309. Arnold, B. C., and Groeneveld, R. A. (1995), “Measuring Skewness With Respect to the Mode,” The American Statistician, 49, 34–38. ArellanoValle, R. B., G´omez, H. W., and Quintana, F. A. (2005), “Statistical Inference for a General Class of Asymmetric Distributions,” Journal of Statistical Planning and Inference, 128, 427–443. Azzalini, A. (1985), “A Class of Distributions Which Includes the Normal Ones,” Scandinavian Journal of Statistics, 12, 171178. Azzalini, A. (1986), “Further Results on a Class of Distributions Which Includes the Normal Ones,” Statistica, 46, 199–208. Azzalini, A., and Capitanio, A. (2003), “Distributions Generated by Perturbation of Symmetry With Emphasis on a Multivariate Skewt Distribution,” Journal of the Royal Statistical Society B, 65, 367– 389. BarndorffNielsen, O. E. (1977), “Exponentially Decreasing Distributions for the Logarithm of Particle Size,” Proceedings of the Royal Society of London A, 353, 401419. Berlaint, J., Goegebeur, Y., Segers, J., and Teugels, J. (2004), Statistics of Extremes: Theory and Applications, Wiley, New York. Christen, J. A., and Fox, C. (2010), “A General Purpose Sampling Algorithm for Continuous Distributions (The twalk),” Bayesian Analysis, 5, 1–20. Critchley, F., and Jones, M. C. (2008), “Asymmetry and Gradient Asymmetry Functions: DensityBased Skewness and Kurtosis,” Scandinavian Journal of Statistics, 35, 415437. Doss, H., and Hobert, J. P. (2010), “Estimation of Bayes Factors in a Class of Hierarchical Random Effects Models Using Geometrically Ergodic MCMC Algorithm,” Journal of Computational and Graphical Statistics, 19, 295–312. Dunson, D. B. (2010), “Nonparametric Bayes Applications to Biostatistics,” In Bayesian Nonparametrics (Hjort, N. L., Holmes, C. .C, M¨uller, P. Walker, S. G. Eds.), pp. 223–273. Cambridge University Press, Cambridge. Fern´andez, C., Osiewalski, J., and Steel, M. F. J. (1995), “Modeling and Inference With vSpherical Distributions,” Journal of the American Statistical Association, 90, 13311340. Fern´andez, C., and Steel, M. F. J. (1998a), “On Bayesian Modeling of Fat Tails and Skewness,” Journal of the American Statistical Association, 93, 359–371. Fern´andez, C. and Steel, M. F. J. (1998b). On the dangers of modelling through continuous distributions: A Bayesian perspective, in Bernardo, J. M., Berger, J. O., Dawid, A. P. and Smith, A. F. M. eds., Bayesian Statistics 6, Oxford University Press (with discussion), pp. 213–238. Fern´andez, C., and Steel, M. F. J. (2000), “Bayesian Regression Analysis With Scale Mixtures of Normals,” Econometric Theory, 16, 80–101. Ferreira, J. T. A. S., and Steel, M. F. J. (2006), “A Constructive Representation of Univariate Skewed Distributions,” Journal of the American Statistical Association, 101, 823–829. Ferreira, J. T. A. S., and Steel, M. F. J. (2007), “A New Class of Skewed Multivariate DistributionsWith Applications to Regression Analysis,” Statistica Sinica, 17, 505–529. Finley, A. O., and Banerjee, S. (2013), “spBayes: Univariate and Multivariate Spatialtemporal Modeling,” R package version 0.38, http://CRAN.Rproject.org/package=spBayes. Fischer, M., and Klein, I. (2004), “Kurtosis Modelling by Means of the J−Transformation,” Allgemeines Statistisches Archiv, 88, 35–50. Goerg, G. M. (2011), “Lambert W Random Variables  A New Generalized Family of Skewed Distributions with Applications to Risk Estimation,” The Annals of Applied Statistics, 5, 2197–2230. Groeneveld, R. A., and Meeden, G. (1984), “Measuring Skewness and Kurtosis,” The Statistician, 33, 391399. Hansen, B. E. (1994), “Autoregressive Conditional Density Estimation,” International Economic Review, 35, 705–730. Haynes, M. A., MacGilllivray, H. L., and Mergersen, K. L. (1997), “Robustness of Ranking and Selection Rules Using Generalized g and k Distributions,” Journal of Statistical Planning and Inference, 65, 45–66. Johnson, N. L. (1949), “Systems of Frequency Curves Generated by Methods of Translation,” Biometrika, 36, 149–176. Jones, M. C. (2014), “On Families of Distributions With Shape Parameters,” International Statistical Review, in press (with discussion). Jones, M. C., and AnayaIzquierdo K. (2010), “On Parameter Orthogonality in Symmetric and Skew Models,” Journal of Statistical Planning and Inference, 141, 758–770. Jones, M. C., and Faddy, M. J. (2003), “A Skew Extension of the tDistribution, With Applications,” Journal of Royal Statistical Society Series B, 65, 159–174. Jones, M. C., and Pewsey A. (2009), “Sinharcsinh Distributions,” Biometrika, 96, 761–780. Klein, I., and Fischer, M. (2006), “Power Kurtosis Transformations: Definition, Properties and Ordering,” Allgemeines Statistisches Archiv, 90, 395–401. Ley, C., and Paindaveine, D. (2010), “Multivariate Skewing Mechanisms: A Unified Perspective Based On the Transformation Approach,” Statistics & Probability Letters, 80, 1685–1694. Marinho, V. C. C., Higgins, J. P. T., Logan, S., and Sheiham, A. (2003), “Fluoride Toothpastes for Preventing Dental Caries in Children and Adolescents (Cochrane Review),” The Cochrane Library, (Issue 4 edn). Wiley: Chichester. McCulloch, M. E., and Neuhaus, J. M. (2011), “Misspecifying the Shape of a Random Effects Distribution: Why Getting It Wrong May Not Matter,” Statistical Science, 26, 388–402. Mudholkar, G. S., and Hutson, A. D. (2000), “The Epsilonskewnormal Distribution for Analyzing Nearnormal Data” Journal of Statistical Planning and Inference, 83, 291–309. Polson, N., and Scott, J. G. (2012), “On the HalfCauchy Prior for a Global Scale Parameter,” Bayesian Analysis, 7, 887–902. RamirezCobo, P., Lillo, R. E., Wilson, S., and Wiper, M. P. (2010), “Bayesian Inference for Double Pareto Lognormal Queues,” The Annals of Applied Statistics, 4, 1533–1557. Reed, W., and Jorgensen, M. (2004), “The double Paretolognormal Distribution – A New Parametric Model for Size Distributions,” Communications in Statistics, Theory & Methods, 33, 17331753. Roberts, G. O., and Rosenthal, J. S. (2009), “Examples of Adaptive MCMC,” Journal of Computational and Graphical Statistics, 18, 349–367. Rosco, J. F., Jones, M. C., and Pewsey, A. (2011), “Skew t Distributions Via the Sinharcsinh Transformation,” TEST 20: 630–652. Rubio, F. J. (2013), Modelling of Kurtosis and Skewness: Bayesian Inference and Distribution Theory, PhD Thesis, University of Warwick, UK. Rubio, F. J., and Steel, M. F. J. (2013), “Bayesian Inference for P(X < Y ) Using Asymmetric Dependent Distributions,” Bayesian Analyisis, 8, 43–62. Rubio, F. J., and Steel, M. F. J. (2014), “Inference in TwoPiece LocationScale models With Jeffreys Priors (with discussion),” Bayesian Analyisis, 9, 1–22. Thompson, S. G., and Lee, K. J. (2008), “Flexible Parametric Models for Random–Effects Distributions,” Statistics in Medicine, 27, 418–434. Tukey, J. M. (1977), Exploratory Data Analysis, AddisonWesley, Reading, M. A. van Zwet, W. R. (1964), Convex Transformations of Random Variables, Mathematisch Centrum, Amsterdam. Venturini, S., Dominici, F., and Parmigiani, G. (2008), “Gamma Shape Mixtures for Heavytailed Distributions,” Annals of Applied Statistics, 2, 756–776. Wang, J., Boyer, J., and Genton M. C. (2004), “A Skew Symmetric Representation of Multivariate Distributions,” Statistica Sinica, 14, 1259–1270. Zhang, D., and Davidian, M. (2001). “Linear Mixed Models with Flexible Distributions of Random Effects for Longitudinal Data,” Biometrics 57, 795–802. Zhu, D., and Galbraith, J. W. (2010), “A Generalized Asymmetric Studentt Distribution With Application to Financial Econometrics,” Journal of Econometrics, 157, 297–305. Zhu, D., and Galbraith, J. W. (2011), “Modeling and Forecasting Expected Shortfall With the Generalized Asymmetric Studentt and Asymmetric Exponential Power Distributions,” Journal of Empirical Finance, 18, 765–778. Zhu, D., and ZindeWalsh, V. (2009), “Properties and Estimation of Asymmetric Exponential Power Distribution,” Journal of Econometrics, 148, 8699. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/57102 