Bayesian Procedures to Nonparametric Hypothesis Testing and Model Selection in High-Dimensional Quantile Regression Models

Bayesian procedures to hypothesis testing and model selection in linear models have received much attention in the literature. In this work, we first considered Bayesian hypothesis testing for the presence of an association, described by Kendall's τ coefficient, between two variables measured on an ordinal scale. Owing to the absence of the likelihood functions for the data, we employed the asymptotic normal distributions of the test statistic as the working likelihoods and derived a simple and easy-to-implement Bayes factor. Investigating the asymptotic behavior of the Bayes factor, we found the conditions of the priors so that it is consistent with whichever the hypothesis is true. We then generalized Bayesian testing procedures based on a wide class of nonparametric statistics having either asymptotic normal distribution or chi-squared distribution. We derived a closed-form Bayes factor for nonparametric hypothesis testing that can be easily computed in standard software and is also consistent with the true hypothesis under some conditions. Recently, with the rapid development of computational technology, the scale of the routinely collected data increases greatly and so does the number of predictors. It may not be suitable to directly implement traditional statistical models to analyze this type of data, since the resulting models could suffer greatly from severe overfitting and thus yield poor predictive performance. In this work, we considered Bayesian model selection in quantile regression models with a bridge-randomized penalization that employs a prior for the penalty index, as opposed to the conventional bridge penalization with a fixed penalty in high-dimensional settings, in which the model dimension could greatly exceed the sample size. We proposed an efficient Bayesian computational algorithm via a two-block Markov Chain Monte Carlo algorithm for exploring the full posterior distributions of the unknown parameters with good convergence and mixing proprieties.