用于稳健贝叶斯分位数回归的对数正则变化尺度混合非对称拉普拉斯分布
Log-regularly varying scale mixture of asymmetric Laplaces for robust Bayesian quantile regression
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中文总结 AI 辅助
该研究针对基于非对称拉普拉斯分布的贝叶斯分位数回归对极端观测值敏感的问题，提出AL-LPAL混合分布，建立后验稳健性并开发后验计算方法，经模拟和实际数据验证其预测性能良好。
中文摘要 AI 辅助
基于非对称拉普拉斯（AL）分布的贝叶斯分位数回归对极端观测值较为敏感，因为其尾部呈指数衰减。我们提出一种稳健误差分布，构造为AL分布与非对称拉普拉斯分布的对数帕累托尺度混合（LPAL）的有限混合。与AL分布的正态位置-尺度表示的直接对数帕累托扩展不同，所提AL-LPAL混合分布保留了规定分位数，且在两个尾部均呈现对数正则变化行为。LPAL分量在目标分位数处还具有无界密度，产生的分布兼具尖锐的中心聚集性与超厚尾部。我们在任意极端污染下建立了后验稳健性，并为回归系数和尺度参数的后验矩存在性提供了充分条件。对于后验计算，我们利用潜变量增广开发了吉布斯采样器，以及计算高效的平均场变分贝叶斯近似。模拟研究表明，所提方法在中度污染下具有竞争力，且在感兴趣分位数受严重污染时，能保持稳定的点估计和相对集中的后验区间。将其应用于二氧化碳数据和波士顿住房数据，采用与现有稳健贝叶斯分位数回归分析相同的预处理，在几乎所有考虑的分位数水平和损失准则下均表现出良好的预测性能。
英文摘要
Bayesian quantile regression based on the asymmetric Laplace (AL) distribution can be sensitive to extreme observations because of its exponentially decaying tails. We propose a robust error distribution constructed as a finite mixture of the AL distribution and a log-Pareto scale mixture of asymmetric Laplace distributions (LPAL). Unlike a direct log-Pareto extension of the normal location-scale representation of the AL distribution, the proposed AL-LPAL mixture preserves the prescribed quantile and exhibits log-regularly varying behavior in both tails. The LPAL component also has an unbounded density at the target quantile, yielding a distribution that combines sharp central concentration with super-heavy tails. We establish posterior robustness under arbitrarily extreme contamination and provide sufficient conditions for the existence of posterior moments of the regression coefficients and scale parameter. For posterior computation, we develop a Gibbs sampler using latent-variable augmentations and a computationally efficient mean-field variational Bayes approximation. Simulation studies show that the proposed method is competitive under moderate contamination and maintains stable point estimation with comparatively concentrated posterior intervals, particularly when severe contamination affects the quantile of interest. Applications to carbon dioxide and Boston housing data, using the same preprocessing as existing robust Bayesian quantile regression analyses, show favorable predictive performance across nearly all quantile levels and loss criteria considered.
