scholarly journals Bayesian sparse linear regression with unknown symmetric error

2019 ◽  
Vol 8 (3) ◽  
pp. 621-653 ◽  
Author(s):  
Minwoo Chae ◽  
Lizhen Lin ◽  
David B Dunson
Keyword(s):  
Linear Regression ◽  
Dirichlet Process ◽  
Regression Coefficients ◽  
Dirichlet Process Mixture ◽  
Model Selection Consistency ◽  
Bayesian Procedures ◽  
Mixing Distributions ◽  
Error Density ◽  
Von Mises ◽  
Mixture Of Gaussian

Abstract We study Bayesian procedures for sparse linear regression when the unknown error distribution is endowed with a non-parametric prior. Specifically, we put a symmetrized Dirichlet process mixture of Gaussian prior on the error density, where the mixing distributions are compactly supported. For the prior on regression coefficients, a mixture of point masses at zero and continuous distributions is considered. Under the assumption that the model is well specified, we study behavior of the posterior with diverging number of predictors. The compatibility and restricted eigenvalue conditions yield the minimax convergence rate of the regression coefficients in $\ell _1$- and $\ell _2$-norms, respectively. In addition, strong model selection consistency and a semi-parametric Bernstein–von Mises theorem are proven under slightly stronger conditions.

scholarly journals Weighted Dirichlet Process Mixture Models to Accommodate Complex Sample Designs for Linear and Quantile Regression

2021 ◽  
Vol 37 (1) ◽  
pp. 71-95
Author(s):  
Michael R. Elliott ◽  
Xi Xia
Keyword(s):  
Linear Regression ◽  
Quantile Regression ◽  
Dirichlet Process ◽  
Predictive Distribution ◽  
Efficient Estimation ◽  
Dirichlet Process Mixture ◽  
Posterior Predictive Distribution ◽  
Health And Nutrition ◽  
Regression Parameters ◽  
The Us

Abstract Standard randomization-based inference conditions on the data in the population and makes inference with respect to the repeating sampling properties of the sampling indicators. In some settings these estimators can be quite unstable; Bayesian model-based approaches focus on the posterior predictive distribution of population quantities, potentially providing a better balance between bias correction and efficiency. Previous work in this area has focused on estimation of means and linear and generalized linear regression parameters; these methods do not allow for a general estimation of distributional functions such as quantile or quantile regression parameters. Here we adapt an extended Dirichlet Process Mixture model that allows the DP prior to be a mixture of DP random basis measures that are a function of covariates. These models allow many mixture components when necessary to accommodate the sample design, but can shrink to few components for more efficient estimation when the data allow. We provide an application to the estimation of relationships between serum dioxin levels and age in the US population, either at the mean level (via linear regression) or across the dioxin distribution (via quantile regression) using the National Health and Nutrition Examination Survey.

A robust random coefficient regression representation of the chain-ladder method

2021 ◽  
pp. 1-32
Author(s):  
Ioannis Badounas ◽  
Apostolos Bozikas ◽  
Georgios Pitselis
Keyword(s):  
Linear Regression ◽  
Linear Regression Model ◽  
Random Coefficients ◽  
Regression Coefficients ◽  
Varying Coefficients ◽  
Statistical Procedures ◽  
Chain Ladder Method ◽  
Loss Reserving ◽  
Random Coefficient Regression ◽  
Chain Ladder

Abstract It is well known that the presence of outliers can mis-estimate (underestimate or overestimate) the overall reserve in the chain-ladder method, when we consider a linear regression model, based on the assumption that the coefficients are fixed and identical from one observation to another. By relaxing the usual regression assumptions and applying a regression with randomly varying coefficients, we have a similar phenomenon, i.e., mis-estimation of the overall reserves. The lack of robustness of loss reserving regression with random coefficients on incremental payment estimators leads to the development of this paper, aiming to apply robust statistical procedures to the loss reserving estimation when regression coefficients are random. Numerical results of the proposed method are illustrated and compared with the results that were obtained by linear regression with fixed coefficients.

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