Abstract
The multivariate skew-t distribution (J Multivar Anal 79:93–113, 2001; J R Stat Soc, Ser B 65:367–389, 2003; Statistics 37:359–363, 2003) includes the Student t, skew-Cauchy and Cauchy distributions as special cases and the normal and skew–normal ones as limiting cases. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis of repeated measures, pretest/post-test data, under multivariate null intercept measurement error model (J Biopharm Stat 13(4):763–771, 2003) where the random errors and the unobserved value of the covariate (latent variable) follows a Student t and skew-t distribution, respectively. The results and methods are numerically illustrated with an example in the field of dentistry.
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References
Aoki R, Bolfarine H, Singer JM (2001) Null intercept measurement error regression models. Test 10: 441–454
Aoki R, Bolfarine H, Singer JM (2002) Asymptotic efficiency of method of moments estimators under null intercept measurement error regression models. Braz J Probab Stat 16: 157–166
Aoki R, Achcar JA, Bolfarine H, Singer JM (2003a) Bayesian analysis of null intercept errors-in-variables regression for pretest/post-test data. J Appl Stat 30(1): 5–14
Aoki R, Bolfarine H, Achcar JA, Leão Pinto D Jr (2003b) Bayesian analysis of a multivariate null intercept error-in-variables regression model. J Biopharm Stat 13(4): 763–771
Arellano-Valle RB, Bolfarine H, Lachos VH (2005) Skew-normal linear mixed models. J Data Sci 3: 415–438
Arnold BC, Beaver RJ (2002) Skewed multivariate models related to hidden truncation and/or selective reporting. Test 11: 7–54
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12: 171–178
Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distribution. J R Stat Soc 61: 579–602
Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on the multivariate skew-t distribution. J R Stat Soc, Ser B 65: 367–389
Azzalini A, Dalla-Valle A (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726
Berkane M, kano Y, Bentler PM (1994) Pseudo maximum likelihood estimation in elliptical theory: effects of misspecification. Comput Stat Data Anal 18: 255–267
Branco M, Dey D (2001) A general class of multivariate skew-elliptical distribution. J Multivar Anal 79: 93–113
Brooks SP (2002) Discussion on the paper by Spiegelhalter, Best, Carlin, and van de Linde (2002) J R Stat Soc, Ser B 64(3):616–618
Carlin BP, Louis TA (2001) Bayes and empirical bayes methods for data analysis essays on item response theory, 2nd edn. Chapman and Hall, New York
Chan LK, Mak TK (1979) On the maximun likelihood estimation of a linear structural relationship when the intercept is known. J Multivar Anal 9: 304–313
Cheng CL, van Ness JW (1999) Statistical regression with measurement error. Arnold, London
Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman and Hall, New York
Fuller WA (1987) Measurement error models. Wiley, New York
Gelman A, Rubin D (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7: 457–511
Genton MG, Loperfido N (2005) Generalized skew-elliptical distributions and their quadratic forms. Ann Inst Stat Math 57: 389–401
Gupta AK (2003) Multivariate skew t-distributions. Statistics 37: 359–363
Hadgu A, Koch G (1999) Application of generalized estimating equations to a dental randomized clinical trial. J Biopharm Stat 9: 161–178
Hill MA, Dixon WJ (1982) Robustness in real life: a study of clinical laboratory data. Biometrics 38: 377–396
Kendall MG, Stuart A (1973) The advanced theory of statistics, vols 1–2, 4th edn. Griffin, London
Lachos VH, Bolfarine H, Arellano-Valle RB, Montenegro LC (2007) Likelihood based inference for multivariate skew-normal regression models. Commun Stat Theory Methods 36: 1769–1786
Lange KL, Little RJ, Taylor J (1989) Robust statistical modelling using the t-distribuion. J Am Stat Assoc 84: 881–896
Lindley DV (1972) Bayesian statistics: a review. SIAM, Philadelphia
Patefield WM (1985) Information from the maximized likelihood function. Biometrics 72(3): 664–668
Sahu SK, Dey DK, Branco MD (2003) A new class of multivariate skew distributions with aplications to Bayesian regression models. Can J Stat 31: 129–150
Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc, B 55(1): 3–23
Spiegelhalter DJ, Best NG, Carlin BP, Linde AV (2002) Bayesian measures of model complexity and fit. J R Stat Soc 64: 583–639
Tan F, Peng H (2006) The slash and the skew-slash Student t distributions (submitted)
Wang J, Genton M (2006) The multivariate skew-slash distribution. J Stat Plan Inference 136: 209–220
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Lachos, V.H., Cancho, V.G. & Aoki, R. Bayesian analysis of skew-t multivariate null intercept measurement error model. Stat Papers 51, 531–545 (2010). https://doi.org/10.1007/s00362-008-0138-z
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DOI: https://doi.org/10.1007/s00362-008-0138-z