Heteroscedastic Double Bayesian Elastic Net
This addresses regression modeling for practitioners dealing with heteroscedastic and high-dimensional data, representing an incremental advancement by extending Bayesian elastic net methods.
The paper tackles the problem of regression with non-constant error variance in high-dimensional settings by proposing the Heteroscedastic Double Bayesian Elastic Net (HDBEN), which jointly models mean and variance with Bayesian priors, achieving improved performance in simulations under heteroscedasticity and high dimensionality.
In many practical applications, regression models are employed to uncover relationships between predictors and a response variable, yet the common assumption of constant error variance is frequently violated. This issue is further compounded in high-dimensional settings where the number of predictors exceeds the sample size, necessitating regularization for effective estimation and variable selection. To address this problem, we propose the Heteroscedastic Double Bayesian Elastic Net (HDBEN), a novel framework that jointly models the mean and log-variance using hierarchical Bayesian priors incorporating both $\ell_1$ and $\ell_2$ penalties. Our approach simultaneously induces sparsity and grouping in the regression coefficients and variance parameters, capturing complex variance structures in the data. Theoretical results demonstrate that proposed HDBEN achieves posterior concentration, variable selection consistency, and asymptotic normality under mild conditions which justifying its behavior. Simulation studies further illustrate that HDBEN outperforms existing methods, particularly in scenarios characterized by heteroscedasticity and high dimensionality.