针对高维数据异方差问题，提出联合建模均值与方差的贝叶斯稀疏方法。
Heteroscedastic Double Bayesian Elastic Net
- 用双层贝叶斯先验同时处理回归系数和方差参数的稀疏性。
- 在异方差和高维条件下，优于现有方法，实现变量选择一致性和渐近正态性。
- 适合处理具有复杂方差结构的高维数据分析，如基因组学、金融建模。
在许多实际应用中，回归模型用于揭示预测变量与响应变量之间的关系，但常假设误差方差恒定，这一假设经常被违反。当预测变量数量超过样本量时，这一问题更加突出，需要通过正则化实现有效估计和变量选择。为此，我们提出异方差双贝叶斯弹性网（HDBEN），一种新框架，通过包含ℓ₁和ℓ₂惩罚的分层贝叶斯先验，联合建模均值和对数方差。该方法同时在回归系数和方差参数中诱导稀疏性和分组效应，捕捉数据中的复杂方差结构。理论结果表明，在温和条件下，所提出的HDBEN可实现后验集中、变量选择一致性及渐近正态性，验证了其行为合理性。模拟研究进一步表明，HDBEN在异方差和高维场景下优于现有方法。
原文摘要 · Abstract (English)
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.
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