重尾预测变量的球余差筛选
Ball-Codifference Screening for Heavy-Tailed Predictors
AI总结：
研究重尾预测变量下的筛选问题，基于扩展余差和球协方差开发球余差，结合随机球几何与余差构建统计量，定义其筛选效用并制定筛选程序，模拟和数据示例表明该方法能提高预测变量恢复能力及回归预测准确性。
AI中文摘要：
高维筛选通常基于协方差、相关性或最小二乘度量构建。当预测变量稀疏或具有重尾分布时，这些汇总度量可能不稳定甚至未定义。基于我们最近关于扩展余差和球协方差的工作，我们开发了球余差用于具有重尾预测变量和响应的统计建模中的边际筛选。所提出的统计量将随机球的秩型几何与基于特征函数构建的余差作为依赖度量相结合，因此无需明确的有限一阶或二阶矩即可计算。我们定义了球余差及其归一化筛选效用，并制定了一种确定性独立筛选程序。在标准非退化和正则条件下，大样本正态性可通过有界V统计量和泛函德尔塔方法论证得出。高斯和次高斯稳定设计下的模拟研究表明，余差加权球筛选在恢复高度相关预测变量方面具有竞争力或有所改进，特别是在尾部较重时。此外，我们的数据示例表明我们的变量筛选方法显著提高了线性回归中的预测准确性。
英文摘要：
High-dimensional screening is commonly built on covariance, correlation, or least-squares measures. These summary measures can be unstable or even undefined, when predictors are sparse or have heavy-tailed distributions. Building on our recent work on extended codifference and the idea of Ball-covariance, we develop Ball-codifference for marginal screening in statistical modeling with heavy-tailed predictors and responses. The proposed statistic combines the rank-type geometry of random balls with the codifference as a dependency measure constructed based on the characteristic function, so it can be computed without requiring well-defined finite first or second moments. We define Ball-codifference and its normalized screening utility, and formulate a sure independence screening procedure. Large-sample normality follows from a bounded V-statistic and functional-delta-method argument under standard nondegeneracy and regularity conditions. Simulation studies under Gaussian and sub-Gaussian stable designs show that codifference-weighted Ball screening gives competitive or improved recovery of highly associated predictors, especially when tail heaviness is pronounced. Also, our data example illustrates that our variable screening method significantly improves prediction accuracy in linear regression.
