Abstract
In this paper, we propose a Bayesian semiparametric mean-covariance regression model with known covariance structures. A mixture model is used to describe the potential non-normal distribution of the regression errors. Moreover, an empirical likelihood adjusted mixture of Dirichlet process model is constructed to produce distributions with given mean and variance constraints. We illustrate through simulation studies that the proposed method provides better estimations in some non-normal cases. We also demonstrate the implementation of our method by analyzing the data set from a sleep deprivation study.
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References
Antoniak, C. E.: Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist., 2, 1152–1174 (1974)
Bates, D., Mächler, M., Bolker, B., et al.: Fitting linear mixed-effects models using lme4. J. Stat. Softw., 67, 1–48 (2015)
Belenky, G., Wesensten, N. J., Thorne, D. R., et al.: Patterns of performance degradation and restoration during sleep restriction and subsequent recovery: A sleep dose-response study. J. Sleep Res., 12, 1–12 (2003)
Blackwell, D., MacQueen, J. B.: Ferguson distributions via Polya urn schemes. Ann. Statist., 1, 353–355 (1973)
Brunner, L. J., Lo, A. Y.: Bayes methods for a symmetric unimodal density and its mode. Ann. Statist., 17, 1550–1566 (1989)
Escobar, M. D.: Estimating normal means with a Dirichlet process prior. J. Amer. Statist. Assoc., 89, 268–277 (1994)
Escobar, M. D., West, M.: Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc., 90, 577–588 (1995)
Ferguson, T. S.: A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209–230 (1973)
Gelman, A.: Prior distributions for variance parameters in hierarchical models (comment on an article by Browne and Draper). Bayesian Anal., 1, 515–533 (2006)
Geweke, J.: Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Technical report, Federal Reserve Bank of Minneapolis, Research Department, 1991
Hoff, P. D.: Constrained nonparametric estimation via mixtures. PhD thesis, Department of Statistics, University of Wisconsin, 2000
Hoff, P. D.: Nonparametric estimation of convex models via mixtures. Ann. Statist., 31, 174–200 (2003)
Johnson, R. A., Wichern, D. W.: Applied Multivariate Statistical Analysis, Upper Saddle River, Pearson Prentice Hall, 2007
Laird, N.: Nonparametric maximum likelihood estimation of a mixing distribution. J. Amer. Statist. Assoc., 73, 805–811 (1978)
Lin, T. I., Wang, W. L.: Bayesian inference in joint modelling of location and scale parameters of the t distribution for longitudinal data. J. Statist. Plann. Inference, 141, 1543–1553 (2011)
Lindsay, B.: The geometry of mixture likelihoods: a general theory. Ann. Statist., 11, 86–94 (1983a)
Lindsay, B.: The geometry of mixture likelihoods, part ii: The exponential family. Ann. Statist., 11, 783–792 (1983b)
Lindsay, B.: Mixture models: theory, geometry and applications. In NSF-CBMS Regional Conference Series in Probability and Statistics, volume 5, pages i–163. Institute of Mathematical Statistics and the American Statistical Association, 1995
Littell, R. C., Pendergast, J., Natarajan, R.: Modelling covariance structure in the analysis of repeated measures data. Stat. Med., 19, 1793–1819 (2000)
MacEachern, S. N., Müller, P.: Estimating mixture of Dirichlet process models. J. Comput. Graph. Statist., 7, 223–238 (1998)
Neal, R. M.: Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Statist., 9, 249–265 (2000)
Owen, A. B.: Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, 237–249 (1988)
Owen, A. B.: Empirical Likelihood, CRC Press, Boca Raton, 2001
Pan, J., MacKenzie, G.: On modelling mean-covariance structures in longitudinal studies. Biometrika, 90, 239–244 (2003)
Pan, J., MacKenzie, G.: Regression models for covariance structures in longitudinal studies. Stat. Model., 6, 43–57 (2006)
Pan, J., MacKenzie, G.: Modelling conditional covariance in the linear mixed model. Stat. Model., 7, 49–71 (2007)
Park, T., Min, S.: Partially collapsed Gibbs sampling for linear mixed-effects models. Comm. Statist. Theory Methods, 45, 165–180 (2016)
Pourahmadi, M.: Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika, 86, 677–690 (1999)
Yang, M., Dunson, D. B., Baird, D.: Semiparametric Bayes hierarchical models with mean and variance constraints. Comput. Statist., 54, 2172–2186 (2010)
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Supported by National Natural Science Foundation of China (Grant Nos. 11171007/A011103, 11171230 and 11471024)
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Yu, H.J., Shen, J.S., Li, Z.N. et al. Semiparametric Bayesian inference for mean-covariance regression models. Acta. Math. Sin.-English Ser. 33, 748–760 (2017). https://doi.org/10.1007/s10114-016-6357-7
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DOI: https://doi.org/10.1007/s10114-016-6357-7