- Volume 27 Issue 2
DOI QR Code
Bayesian inference for an ordered multiple linear regression with skew normal errors
- Jeong, Jeongmun (Department of Statistics, Pusan National University) ;
- Chung, Younshik (Department of Statistics, Pusan National University)
- Received : 2019.07.13
- Accepted : 2019.12.18
- Published : 2020.03.31
This paper studies a Bayesian ordered multiple linear regression model with skew normal error. It is reasonable that the kind of inherent information available in an applied regression requires some constraints on the coefficients to be estimated. In addition, the assumption of normality of the errors is sometimes not appropriate in the real data. Therefore, to explain such situations more flexibly, we use the skew-normal distribution given by Sahu et al. (The Canadian Journal of Statistics, 31, 129-150, 2003) for error-terms including normal distribution. For Bayesian methodology, the Markov chain Monte Carlo method is employed to resolve complicated integration problems. Also, under the improper priors, the propriety of the associated posterior density is shown. Our Bayesian proposed model is applied to NZAPB's apple data. For model comparison between the skew normal error model and the normal error model, we use the Bayes factor and deviance information criterion given by Spiegelhalter et al. (Journal of the Royal Statistical Society Series B (Statistical Methodology), 64, 583-639, 2002). We also consider the problem of detecting an influential point concerning skewness using Bayes factors. Finally, concluding remarks are discussed.
- Azzalini A (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178.
- Azzalini A and Capitanio A (1999). Statistical applications of the multivariate skew-normal distributions, Journal of the Royal Statistical Society Series B (Statistical Methodology), 61, 579-602. https://doi.org/10.1111/1467-9868.00194
- Azzalini A and Dalla-Valle A (1996). The multivariate skew-normal distribution, Biometrika, 83, 715-726. https://doi.org/10.1093/biomet/83.4.715
- Chen MH and Deely JJ (1996). Bayesian analysis for a constrained linear multiple regression problem for predicting the new crop of apples, Journal of Agricultural, Biological, and Environmental Statistics, 1, 467-489. https://doi.org/10.2307/1400440
- Chib B and Greenberg E (1995). Understanding the metropolis-hastings algorithm, The American Statistician, 49, 327-335.
- Gelman A and Rubin DB (1992). Inference from iterative simulation using multiple sequences, Statistical Science, 7, 457-472. https://doi.org/10.1214/ss/1177011136
- Jang J, Chung Y, Kim C, and Song S (2009). Bayesian meta-analysis using skewed elliptical distributions, Journal of Statistical Computation and Simulation, 79, 691-704. https://doi.org/10.1080/00949650801891595
- Jung M, Song S, and Chung Y (2018). Bayesian change-point problem using Bayes factor with hierarchical prior distribution, Communications in Statistics - Theory and Methods, 46, 1352-1366.
- Newton MA and Raftery AE (1994). Approximate Bayesian inference with the weighted likelihood bootstrap (with discussion), Journal of the Royal Statistical Society. Series B (Methodology), 56, 3-48.
- O'Hagan A and Leonard T (1976). Bayes estimation subject to uncertainty about parameter constraints, Biometrika, 63, 201-203. https://doi.org/10.1093/biomet/63.1.201
- Pettit LI and Young KDS (1990). Measuring the effect of observations on Bayes factors, Biometrika, 77, 455-466. https://doi.org/10.1093/biomet/77.3.455
- Robert CP (1995). Simulation of truncated normal variables, Statistics and Computing, 5, 121-125. https://doi.org/10.1007/BF00143942
- Sahu SK, Dey DK, and Branco MD (2003). A new class of multivariate skew distributions with applications to Bayesian regression models, The Canadian Journal of Statistics, 31, 129-150. https://doi.org/10.2307/3316064
- Spiegelhalter DJ, Best NG, Carlin BP, and Van der Linde A (2002). Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society Series B (Statistical Methodology), 64, 583-639. https://doi.org/10.1111/1467-9868.00353
- Tanner MA and Wong WH (1987). The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association, 82, 528-540. https://doi.org/10.1080/01621459.1987.10478458