## Electronic Journal of Statistics

### Bayesian modelling of skewness and kurtosis with Two-Piece Scale and shape distributions

#### Abstract

We formalise and generalise the definition of 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. Four-parameter subfamilies of this class of distributions that capture different types of asymmetry are discussed. We propose interpretable scale and location-invariant benchmark priors and derive conditions for the propriety 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 data from finance, internet traffic and medicine, comparing them with appropriate competitors.

#### Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 1884-1912.

Dates
First available in Project Euclid: 27 August 2015

https://projecteuclid.org/euclid.ejs/1440680330

Digital Object Identifier
doi:10.1214/15-EJS1060

Mathematical Reviews number (MathSciNet)
MR3391123

Zentralblatt MATH identifier
1331.62090

#### Citation

Rubio, F. J.; Steel, M. F. J. Bayesian modelling of skewness and kurtosis with Two-Piece Scale and shape distributions. Electron. J. Statist. 9 (2015), no. 2, 1884--1912. doi:10.1214/15-EJS1060. https://projecteuclid.org/euclid.ejs/1440680330

#### 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 Beaver, R. J. (2002), “Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion),”, Test, 11, 7–54.
• Arnold, B. C. and Groeneveld, R. A. (1995), “Measuring skewness with respect to the mode,”, The American Statistician, 49, 34–38.
• Arellano-Valle, R. B., Gómez, 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, 171–178.
• 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 skew-$t$ distribution,”, Journal of the Royal Statistical Society B, 65, 367–389.
• Barndorff-Nielsen, O., Kent, J., and Sørensen, M. (1982), “Normal variance-mean mixtures and $z$ distributions,”, International Statistical Review, 145–159.
• 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 $t$-walk),”, Bayesian Analysis, 5, 1–20.
• Critchley, F. and Jones, M. C. (2008), “Asymmetry and gradient asymmetry functions: Density-based skewness and kurtosis,”, Scandinavian Journal of Statistics, 35, 415–437.
• 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., Müller, P., and Walker, S. G., Eds.), pp. 223–273. Cambridge University Press, Cambridge.
• Fernández, C., Osiewalski, J., and Steel, M. F. J. (1995), “Modeling and inference with $v$-spherical distributions,”, Journal of the American Statistical Association, 90, 1331–1340.
• Fernández, C. and Steel, M. F. J. (1998a), “On Bayesian modeling of fat tails and skewness,”, Journal of the American Statistical Association, 93, 359–371.
• Fernández, C. and Steel, M. F. J. (1998b), “On the dangers of modelling through continuous distributions: A Bayesian perspective”, in:, Bayesian Statistics 6 (Bernardo, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M., Eds.), pp. 213–238. Oxford University Press (with discussion).
• Fernández, 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 distributions with applications to regression analysis,”, Statistica Sinica, 17, 505–529.
• Finley, A. O., Banerjee, S., and Carlin, B. P. (2007), “spBayes: An R package for univariate and multivariate hierarchical point-referenced spatial models,”, Journal of Statistical Software, 19, 1–24.
• Fischer, M. and Klein, I. (2004), “Kurtosis modelling by means of the $J$-transformation,”, Allgemeines Statistisches Archiv, 88, 35–50.
• Fonseca, T., Ferreira, M., and Migon, H. (2008), “Objective Bayesian analysis for the student-$t$ regression model,”, Biometrika, 95, 325–333.
• 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, 391–399.
• 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.
• Hoaglin, D. C., Mosteller, F., and Tukey, J. W. (1985), Exploring Data Table, Trends, and Shapes, Wiley, New York.
• Johnson, N. L. (1949), “Systems of frequency curves generated by methods of translation,”, Biometrika, 36, 149–176.
• Jones, M. C. (2014a), “Generating distributions by transformation of scale,”, Statistica Sinica, in press.
• Jones, M. C. (2014b), “On families of distributions with shape parameters (with discussion),”, International Statistical Review, in press.
• Jones, M. C., and Anaya-Izquierdo, 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 $t$-distribution, with applications,”, Journal of Royal Statistical Society, Series B, 65, 159–174.
• Jones, M. C. and Pewsey, A. (2009), “Sinh-arcsinh distributions,”, Biometrika, 96, 761–780.
• Juárez, M. A. and Steel, M. F. J. (2010), “Non-Gaussian dynamic Bayesian modelling for panel data,”, Journal of Applied Econometrics, 25, 1128–1154.
• Klein, I. and Fischer, M. (2006), “Power kurtosis transformations: Definition, properties and ordering,”, Allgemeines Statistisches Archiv, 90, 395–401.
• Ley, C. (2015), “Flexible modelling in statistics: Past, present and future,”, Journal de la Société Française de Statistique, 156, 76–96.
• 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),” in:, 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 epsilon-skew-normal distribution for analyzing near-normal data,”, Journal of Statistical Planning and Inference, 83, 291–309.
• Polson, N. and Scott, J. G. (2012), “On the half-Cauchy prior for a global scale parameter,”, Bayesian Analysis, 7, 887–902.
• Quintana, F. A., Steel, M. F. J., and Ferreira, J. T. A. S. (2009), “Flexible univariate continuous distributions,”, Bayesian Analysis, 4, 497–522.
• Ramirez-Cobo, 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 Pareto–lognormal distribution – a new parametric model for size distributions,”, Communications in Statistics, Theory & Methods, 33, 1733–1753.
• 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 sinh-arcsinh 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. (2014), “Letter to the editor: On the use of improper priors for the shape parameters of asymmetric exponential power models,”, Statistics and Computing, in press.
• Rubio, F. J., Ogundimu, E. O., and Hutton, J. L. (2015), “On modelling asymmetric data using two–piece sinh–arcsinh distributions,”, Brazilian Journal of Probability and Statistics, in press.
• Rubio, F. J. and Steel, M. F. J. (2013), “Bayesian inference for $\mboxP(X<Y)$ using asymmetric dependent distributions,”, Bayesian Analysis, 8, 43–62.
• Rubio, F. J. and Steel, M. F. J. (2014), “Inference in two-piece location-scale models with Jeffreys priors (with discussion),”, Bayesian Analysis, 9, 1–22.
• Thompson, S. G. and Lee, K. J. (2008), “Flexible parametric models for random–effects distributions,”, Statistics in Medicine, 27, 418–434.
• 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 heavy-tailed distributions,”, Annals of Applied Statistics, 2, 756–776.
• Villa, C. and Walker, S. G. (2014), “Objective prior for the number of degrees of freedom of a $t$ distribution,”, Bayesian Analysis, 9, 197–220.
• 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 student-$t$ 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 student-$t$ and asymmetric exponential power distributions,”, Journal of Empirical Finance, 18, 765–778.
• Zhu, D. and Zinde-Walsh, V. (2009), “Properties and estimation of asymmetric exponential power distribution,”, Journal of Econometrics, 148, 86–99.