The Bayesian Regularized Quantile Varying Coefficient Model

The quantile varying coefficient (VC) model can flexibly capture dynamical patterns of regression coefficients. In addition, due to the quantile check loss function, it is robust against outliers and heavy-tailed distributions of the response variable, and can provide a more comprehensive picture of modeling via exploring the conditional quantiles of the response variable. Although extensive studies have been conducted to examine variable selection for the high-dimensional quantile varying coefficient models, the Bayesian analysis has been rarely developed. The Bayesian regularized quantile varying coefficient model has been proposed to incorporate robustness against data heterogeneity while accommodating the non-linear interactions between the effect modifier and predictors. Selecting important varying coefficients can be achieved through Bayesian variable selection. Incorporating the multivariate spike-and-slab priors further improves performance by inducing exact sparsity. The Gibbs sampler has been derived to conduct efficient posterior inference of the sparse Bayesian quantile VC model through Markov chain Monte Carlo (MCMC). The merit of the proposed model in selection and estimation accuracy over the alternatives has been systematically investigated in simulation under specific quantile levels and multiple heavy-tailed model errors. In the case study, the proposed model leads to identification of biologically sensible markers in a non-linear gene-environment interaction study using the NHS data

Modeling operational risk data reported above a time-varying threshold

Typically, operational risk losses are reported above a threshold. Fitting data reported above a constant threshold is a well known and studied problem. However, in practice, the losses are scaled for business and other factors before the fitting and thus the threshold is varying across the scaled data sample. A reporting level may also change when a bank changes its reporting policy. We present both the maximum likelihood and Bayesian Markov chain Monte Carlo approaches to fitting the frequency and severity loss distributions using data in the case of a time varying threshold. Estimation of the annual loss distribution accounting for parameter uncertainty is also presented

Bayesian adaptive lasso quantile regression

Recently, variable selection by penalized likelihood has attracted much research interest. In this paper, we propose adaptive Lasso quantile regression (BALQR) from a Bayesian perspective. The method extends the Bayesian Lasso quantile regression by allowing different penalization parameters for different regression coefficients. Inverse gamma prior distributions are placed on the penalty parameters. We treat the hyperparameters of the inverse gamma prior as unknowns and estimate them along with the other parameters. A Gibbs sampler is developed to simulate the parameters from the posterior distributions. Through simulation studies and analysis of a prostate cancer dataset, we compare the performance of the BALQR method proposed with six existing Bayesian and non-Bayesian methods. The simulation studies and the prostate cancer data analysis indicate that the BALQR method performs well in comparison to the other approaches

Bayesian Tobit quantile regression using-prior distribution with ridge parameter

A Bayesian approach is proposed for coefficient estimation in the Tobit quantile regression model. The proposed approach is based on placing a g-prior distribution depends on the quantile level on the regression coefficients. The prior is generalized by introducing a ridge parameter to address important challenges that may arise with censored data, such as multicollinearity and overfitting problems. Then, a stochastic search variable selection approach is proposed for Tobit quantile regression model based on g-prior. An expression for the hyperparameter g is proposed to calibrate the modified g-prior with a ridge parameter to the corresponding g-prior. Some possible extensions of the proposed approach are discussed, including the continuous and binary responses in quantile regression. The methods are illustrated using several simulation studies and a microarray study. The simulation studies and the microarray study indicate that the proposed approach performs well

Bayesian computation via empirical likelihood

Approximate Bayesian computation (ABC) has become an essential tool for the analysis of complex stochastic models when the likelihood function is numerically unavailable. However, the well-established statistical method of empirical likelihood provides another route to such settings that bypasses simulations from the model and the choices of the ABC parameters (summary statistics, distance, tolerance), while being convergent in the number of observations. Furthermore, bypassing model simulations may lead to significant time savings in complex models, for instance those found in population genetics. The BCel algorithm we develop in this paper also provides an evaluation of its own performance through an associated effective sample size. The method is illustrated using several examples, including estimation of standard distributions, time series, and population genetics models.Comment: 21 pages, 12 figures, revised version of the previous version with a new titl