Tractable Bayesian Variable Selection: Beyond Normality

References

• Alfons, A., Croux, C., and Gelper, S. (2013), “Sparse Least Trimmed Squares Regression for Analyzing High-Dimensional Large Data Sets,” The Annals of Applied Statistics, 7, 226–248.
   Web of Science ®Google Scholar
• Arellano-Valle, R., Gómez, H., and Quintana, F. (2005), “Statistical Inference for a General Class of Asymmetric Distributions,” Journal of Statistical Planning and Inference, 128, 427–443.
   Web of Science ®Google Scholar
• Arslan, O. (2012), “Weighted LAD-LASSO Method for Robust Parameter Estimation and Variable Selection in Regression,” Computational Statistics and Data Analysis, 56, 1952–1965.
   Web of Science ®Google Scholar
• Barbieri, M., and Berger, J. (2004), “Optimal Predictive Model Selection,” The Annals of Statistics, 32, 870–897.
   Web of Science ®Google Scholar
• Bayarri, M., Berger, J., Forte, A., and Garcia-Donato, G. (2012), “Criteria for Bayesian Model Choice with Application to Variable Selection,” The Annals of statistics, 40, 1550–1577.
   Web of Science ®Google Scholar
• Breheny, P., and Huang, J. (2011), “Coordinate Descent Algorithms for Nonconvex Penalized Regression, with Applications to Biological Feature Selection,” Annals of Applied Statistics, 5, 232–253.
   PubMed Web of Science ®Google Scholar
• Calon, A., Espinet, E., Palomo-Ponce, S., Tauriello, D., Iglesias, M., Céspedes, M., Sevillano, M., Nadal, C., Jung, P., Zhang, X.-F., Byrom, D., Riera, A., Rossell, D., Mangues, R., Massague, J., Sancho, E., and Batlle, E. (2012), “Dependency of Colorectal Cancer on a TGF-Beta-Driven Programme in Stromal Cells for Metastasis Initiation,” Cancer Cell, 22, 571–584.
   PubMed Web of Science ®Google Scholar
• Chae, M., Lin, L., and Dunson, D. (2016), “Bayesian Sparse Linear Regression with Unknown Symmetric Error,” arXiv, 1608.02143, 1–34.
   Google Scholar
• Chung, Y., and Dunson, D. (2009), “Nonparametric Bayes Conditional Distribution Modeling with Variable Selection,” Journal of the American Statistical Association, 104, 1646–1660.
   PubMed Web of Science ®Google Scholar
• Daye, Z., Chen, J., and Li, H. (2012), “High-Dimensional Heteroscedastic Regression with an Application to eQTL Data Analysis,” Biometrics, 68, 316–326.
   PubMed Web of Science ®Google Scholar
• Dette, H., Ley, C., and Rubio, F. (2017), “Natural (non-) Informative Priors for Skew-Symmetric Distributions,” Scandinavian Journal of Statistics. arXiv:1605.02880.
   Google Scholar
• Eicker, F. (1964), “Asymptotic Normality and Consistency of the Least Squares Estimator for Families of Linear Regressions,” Annals of Mathematical Statistics, 34, 447–456.
   Google Scholar
• Fan, J., Fan, Y., and Barut, E. (2014), “Adaptive Robust Variable Selection,” The Annals of Applied Statistics, 42, 324–351.
   Web of Science ®Google Scholar
• Fan, J., and Li, R. (2001), “Variable Selection Via Nonconcave Penalized Likelihood and its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360.
   Web of Science ®Google Scholar
• Fernández, C., and Steel, M. (1998), “On Bayesian Modeling of Fat Tails and Skewness,” Journal of the American Statistical Association, 93, 359–371.
   Web of Science ®Google Scholar
• Gijbels, I., and Vrinssen, I. (2015), “Robust Nonnegative Garrote Variable Selection in Linear Regression,” Computational Statistics and Data Analysis, 85, 1–22.
   Web of Science ®Google Scholar
• Gottardo, R., and Raftery, A. (2007), “Bayesian Robust Transformation and Variable Selection: A Unified Approach,” The Canadian Journal of Statistics, 37, 361–380.
   Web of Science ®Google Scholar
• Grünwald, P., and van Ommen, T. (2017), “Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing it,” Bayesian Analysis, 12, 1069–1103.
   Web of Science ®Google Scholar
• Huber, P. (1973), “Robust Regression: Asymptotics, Conjectures and Monte Carlo,” The Annals of Statistics, 1, 799–821.
   Web of Science ®Google Scholar
• Johnson, V. (2013), “On Numerical Aspects of Bayesian Model Selection in High and Ultrahigh-Dimensional Settings,” Bayesian Analysis, 8, 741–758.
   PubMed Web of Science ®Google Scholar
• Johnson, V., and Rossell, D. (2010), “Prior Densities for Default Bayesian Hypothesis Tests,” Journal of the Royal Statistical Society, Series B, 72, 143–170.
   Google Scholar
• ——— (2012), “Bayesian Model Selection in High-Dimensional Settings,” Journal of the American Statistical Association, 24, 649–660.
   Google Scholar
• Kimber, A. C. (1985), “Methods for the Two-Piece Normal Distribution,” Communications in Statistics—Theory and Methods, 14, 235–245.
   Web of Science ®Google Scholar
• Knight, K. (1999), “Asymptotics for L1-estimators of Regression Parameters Under Heteroscedasticity,” The Canadian Journal of Statistics, 27, 497–507.
   Web of Science ®Google Scholar
• Kocherginsky, M., He, X., and Mu, Y. (2005), “Practical Confidence Intervals for Regression Quantiles,” Journal of Computational and Graphical Statistics, 14, 41–55.
   Web of Science ®Google Scholar
• Koenker, R. (1994), “Confidence Intervals for Regression Quantiles,” in Proceedings of the 5th Prague Symposium on Asymptotic Statistics, Springer-Verlag, pp. 349–359.
   Google Scholar
• ——— (2005), Quantile Regression, Cambridge: Cambridge University Press.
   Google Scholar
• Koenker, R., and Bassett, G. (1982), “Tests of Linear Hypotheses and L1 Estimation,” Econometrica, 50, 1577–1584.
   Web of Science ®Google Scholar
• Kundu, S., and Dunson, D. (2014), “Bayes Variable Selection in Semiparametric Linear Models,” Journal of the American Statistical Association, 109, 437–447.
   PubMed Web of Science ®Google Scholar
• Lambert-Lacroix, S. (2011), “Robust Regression through the Huber’s Criterion and Adaptive Lasso Penalty,” Electronic Journal of Statistics, 5, 1015–1053.
   Web of Science ®Google Scholar
• Levenberg, K. (1944), “A Method for the Solution of Certain Non-Linear Problems in Least Squares,” Quarterly of Applied Mathematics, 2, 164–168.
   Google Scholar
• Loh, P.-L. (2017), “Statistical Consistency and Asymptotic Normality for High-Dimensional Robust M-Estimators,” The Annals of Statistics, 45, 866–896.
   Web of Science ®Google Scholar
• Mallick, H., and Nengjun, Y. (2013), “Bayesian Methods for High Dimensional Linear Models,” Journal of Biometrics & Biostatistics, 1, 005.
   PubMedGoogle Scholar
• Marquardt, D. (1963), “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” SIAM Journal on Applied Mathematics, 11, 431–441.
   Web of Science ®Google Scholar
• Mendelson, S. (2014), “Learning Without Concentration for General Loss Functions,” ArXiv, 1410.3192.
   Google Scholar
• Mudholkar, G., and Hutson, A. (2000), “The Epsilon-Skew-Normal Distribution for Analyzing Near-Normal Data,” Journal of Statistical Planning and Inference, 83, 291–309.
   Web of Science ®Google Scholar
• Pollard, D. (1991), “Asymptotics for least Absolute Deviation Regression Estimators,” Econometric Theory, 7, 186–199.
   Web of Science ®Google Scholar
• Rossell, D., Cook, J., Telesca, D., and Roebuck, P. (2016), mombf: Moment and Inverse Moment Bayes Factors (R Package, Version 1.8.1). Vienna, Austria: R Foundation for Statistical Computing.
   Google Scholar
• Rossell, D., and Telesca, D. (2017), “Non-Local Priors for High-Dimensional Estimation,” Journal of the American Statistical Association, 112, 254–265.
   PubMed Web of Science ®Google Scholar
• Rossell, D., Telesca, D., and Johnson, V. (2013), “High-Dimensional Bayesian Classifiers using Non-Local Priors,” in Statistical Models for Data Analysis XV, eds. P. Giudici, S. Ingrassia, M. Vichi, New York: Springer, pp. 305–314.
   Google Scholar
• Rubio, F., and Genton, M. (2016), “Bayesian Linear Regression with Skew-Symmetric Error Distributions with Applications to Survival Analysis,” Statistics in Medicine, 35, 2441–2454.
   PubMed Web of Science ®Google Scholar
• Rubio, F., and Steel, M. (2014), “Inference in Two-Piece Location-Scale Models with Jeffreys Priors” (with discussion), Bayesian Analysis, 9, 1–22.
   Web of Science ®Google Scholar
• Rubio, F., and Yu, K. (2017), “Flexible Objective Bayesian Linear Regression With Applications in Survival Analysis,” Journal of Applied Statistics, 44, 798–810.
   Web of Science ®Google Scholar
• Scott, J., and Berger, J. (2010), “Bayes and Empirical Bayes Multiplicity Adjustment in the Variable Selection Problem,” The Annals of Statistics, 38, 2587–2619.
   Web of Science ®Google Scholar
• Shin, M., Bhattacharya, A., and Johnson, V. (2018), “Scalable Bayesian Variable Selection using Nonlocal Prior Densities in Ultrahigh-Dimensional Settings,” Statistica Sinica, 28, 1053–1078.
   Google Scholar
• Sorensen, D. (1982), “Newton’s Method with a Model Trust Region Modification,” SIAM Journal of Numerical Analysis, 19, 409–426.
   Web of Science ®Google Scholar
• Srivastava, M. (1971), “On Fixed Width Confidence Bounds for Regression Parameters,” The Annals of Mathematical Statistics, 42, 1403–1411.
   Google Scholar
• Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.
   Google Scholar
• Wallis, K. (2014), “The Two-Piece Normal, Binormal, or Double Gaussian Distribution: Its Origin and Rediscoveries,” Statistical Science, 29, 106–112.
   Web of Science ®Google Scholar
• Wang, H., Li, G., and Jiang, G. (2007), “Robust Regression Shrinkage and Consistent Variable Selection through the LAD-LASSO,” Journal of Business & Economic Statistics, 25, 347–355.
   Web of Science ®Google Scholar
• Wang, L., and Li, R. (2009), “Weighted Wilcoxon-Type Smoothly Clipped Absolute Deviation Method,” Biometrics, 65, 564–571.
   PubMed Web of Science ®Google Scholar
• Wang, L., Tang, Y., Sinha, D., Pati, D., and Lipsitz, S. (2016), “Bayesian Variable Selection for Skewed Heteroscedastic Response,” arXiv preprint arXiv:1602.09100.
   Google Scholar
• Wu, L., and Liu, Y. (2009), “Variable Selection in Quantile Regression,” Statistica Sinica, 19, 801–809.
   Web of Science ®Google Scholar
• Yan, Y., and Kottas, A. (2017), “A New Family of Error Distributions for Bayesian Quantile Regression,” arXiv preprint arXiv:1701.05666.
   Google Scholar
• Yu, K., Chen, C., Reed, C., and Dunson, D. (2013), “Bayesian Variable Selection in Quantile Regression,” Statistics and its Interface, 6, 261–274.
   Web of Science ®Google Scholar
• Yuan, T., Huang, X., Woodcock, M., Du, M., Dittmar, R., Wang, Y., Tsai, S., Kohli, M., Boardman, L., Patel, T., and Wang, L. (2016), “Plasma Extracellular RNA Profiles in Healthy and Cancer Patients,” Scientific Reports, 6, 1–11.
   PubMed Web of Science ®Google Scholar