Advertisement

A Bayesian Analysis of Two-Piece Distributions Based on the Scale Mixtures of Normal Family

  • Behjat Moravveji
  • Zahra Khodadadi
  • Mohsen Maleki
Research Paper
  • 35 Downloads

Abstract

The current paper seeks to present a Bayesian approach for the estimation of the parameters of the two-piece scale mixtures of normal distributions. This is a rich family of light/heavy-tailed symmetric/asymmetric distributions that includes, as a special case, the heavy-tailed scale mixtures of normal distributions, and is flexible in computations for modeling symmetric and asymmetric data. A Bayesian approach is possible from the specification of hierarchical representations of the proposed family. We illustrate the usefulness of our approach with both real and simulated data.

Keywords

Bayesian estimates Informative prior Scale mixtures of normal family Skewness Two-piece distributions 

Notes

Acknowledgements

The authors would like to thank the Associated Editor and four anonymous reviewers for their suggestions, corrections, and encouragement, which helped us to improve earlier versions of the manuscript.

References

  1. Andrews DR, Mallows CL (1974) Scale mixture of normal distribution. J R Stat Soc Series B 36:99–102MathSciNetzbMATHGoogle Scholar
  2. Arellano-Valle RB, Gómez H, Quintana FA (2005) Statistical inference for a general class of asymmetric distributions. J Stat Plan Infer 128:427–443MathSciNetCrossRefzbMATHGoogle Scholar
  3. Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178MathSciNetzbMATHGoogle Scholar
  4. Azzalini A, Arellano-Valle RB (2013) Maximum penalized likelihood estimation for skew-normal and skew t distributions. J Stat Plan Infer 143:419–433MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bondon P (2009) Estimation of autoregressive models with epsilon-skew-normal innovations. J Multivar Anal 100:1761–1776MathSciNetCrossRefzbMATHGoogle Scholar
  6. Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79:99–113MathSciNetCrossRefzbMATHGoogle Scholar
  7. Brooks SP (2002) Discussion on the paper by Spiegelhalter. Best Carlin van der Linde 64(4):616–618Google Scholar
  8. Brooks SP, Gelman A (1998) Alternative methods for monitoring convergence of iterative simulations. J Comput Graph Stat 7(4):434–455Google Scholar
  9. Carlin BP, Louis TA (2001) Bayes and empirical bayes methods for data analysis, 2nd edn. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  10. Contreras-Reyes JE (2016) Analyzing fish condition factor index through skew-gaussian information theory quantifiers. Fluct Noise Lett 15(2):1650013CrossRefGoogle Scholar
  11. Eling M (2012) Fitting insurance claims to skewed distributions: are the skew-normal and skew-student good models? insurance. Math Econ 51:239–248MathSciNetCrossRefGoogle Scholar
  12. Fechner GT (1897) Kollectivmasslehre. Engleman, LeipzigGoogle Scholar
  13. Gelman A, Rubin D (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7:457–511CrossRefzbMATHGoogle Scholar
  14. Gibbons JF, Mylroie S (1973) Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions. Appl Phys Lett 22:568–572CrossRefGoogle Scholar
  15. Gómez HW, Torres FJ, Bolfarine H (2007) Large-sample inference for the epsilon-skew-t distribution. Commun Stat Theor Methods 36(1):73–81MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hutson A (2004) Utilizing the flexibility of the epsilon-skew-normal distribution for common regression problems. J Appl Stat 31(6):673–683MathSciNetCrossRefGoogle Scholar
  17. John S (1982) The three-parameter two-piece normal family of distributions and its fitting. Commun Stat Theor Method 11:879–885MathSciNetCrossRefGoogle Scholar
  18. Kimber AC (1985) Methods for the two-piece normal distribution. Commun Stat Theor Method 14:235–245MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kimber AC, Jeynes C (1987) An application of the truncated two-piece normal distribution to the measurement of depths of arsenic implants in silicon. Appl Stat 36:352–357CrossRefGoogle Scholar
  20. Liseo B (1990) The skew-normal class of densities: inferential aspects from a Bayesian viewpoint (Italian). Statistica 50:71–82zbMATHGoogle Scholar
  21. Liseo B, Loperfido N (2006) A note on reference priors for the scalar skew-normal distribution. J Stat Plan Infer 136(2):373–389MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lopez Quintero FO, Contreras-Reyes JE, Wi R, Arellano-Valle RB (2017) Flexible Bayesian analysis of the von Bertalanffy growth function with the use of a log-skew-t distribution. Fish Bull 115(1):13–26CrossRefGoogle Scholar
  23. Maleki M, Arellano-Valle RB (2016) Maximum a posteriori estimation of autoregressive processes based on finite mixtures of scale-mixtures of skew-normal distributions. J Stat Comp Sim 87(6):1061–1083MathSciNetCrossRefGoogle Scholar
  24. Maleki M, Mahmoudi MR (2017) Two-Piece Location-Scale Distributions based on Scale Mixtures of Normal family. Commun Stat Theor Meth 46(24):12356–12369MathSciNetCrossRefzbMATHGoogle Scholar
  25. Maleki M, Nematollahi AR (2016) Bayesian approach to epsilon-skew-normal family. Commun Stat Theor Method 46:7546–7561MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mudholkar GS, Hutson AD (2000) The epsilon-skew-normal distribution for analyzing near-normal data. J Stat Plan Infer 83:291–309MathSciNetCrossRefzbMATHGoogle Scholar
  27. Rezaie J, Eidsvik J, Mukerji T (2014) Value of information analysis and Bayesian inversion for closed skew-normal distributions: applications to seismic amplitude variation with offset data. Geophysics 79:R151–R163CrossRefGoogle Scholar
  28. Rosa GJM, Padovani CR, Gianola D (2003) Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation. Biom l. 45:573–590MathSciNetGoogle Scholar
  29. Rubio FJ, Steel MFG (2014) Inference in Two-Piece Location-Scale Models with Jeffreys Priors. Bayesian Anal 9(1):1–22MathSciNetCrossRefzbMATHGoogle Scholar
  30. Runnenberg JT (1978) Mean, median, mode. Stat Neerlandica 32:73–79MathSciNetCrossRefGoogle Scholar
  31. Toth Z, Szentimrey T (1990) The bi-normal distribution: a distribution for representing asymmetrical but normal-like weather elements. J Clim 3:128–136CrossRefGoogle Scholar
  32. Wichitaksorn N, Choy ST, Gerlach R (2014) A generalized class of skew distributions and associated robust quantile regression models. Can J Stat 42(4):579–596MathSciNetCrossRefzbMATHGoogle Scholar
  33. Zhou J, Wang X (2008) Accurate closed-form approximation for pricing Asian and basket options. Appl Stoch Model Bus Ind 24:343–358MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shiraz University 2018

Authors and Affiliations

  1. 1.Department of Statistics, Marvdasht BranchIslamic Azad UniversityMarvdashtIran
  2. 2.Department of Statistics, College of SciencesShiraz UniversityShirazIran

Personalised recommendations