The Sorted L-One Estimator (SLOPE) is a popular regularization method in regression, which induces clustering of the estimated coefficients. That is, the estimator can have coefficients of identical magnitude. In this paper, we derive an asymptotic distribution of SLOPE for the ordinary least squares, Huber, and Quantile loss functions, and use it to study the clustering behavior in the limit. This requires a stronger type of convergence since clustering properties do not follow merely from the classical weak convergence. For this aim, we utilize the Hausdorff distance, which provides a suitable notion of convergence for the penalty subdifferentials and a bridge toward weak convergence of the clustering pattern. We establish asymptotic control of the false discovery rate for the asymptotic orthogonal design of the regressor. We also show how to extend the framework to a broader class of regularizers other than SLOPE.

翻译：排序L1估计量（SLOPE）是一种流行的回归正则化方法，可诱导估计系数的聚类效应，即估计量能产生等模值的系数。本文推导了普通最小二乘、Huber和分位数损失函数下SLOPE的渐近分布，并利用该分布研究极限聚类行为。由于聚类性质无法仅通过经典弱收敛获得，因此需要更强的收敛类型。为此，我们采用Hausdorff距离为惩罚次微分提供合适的收敛概念，并搭建通往聚类模式弱收敛的桥梁。我们建立了回归变量渐近正交设计下错误发现率的渐近控制，同时展示了如何将本框架推广至SLOPE之外更广泛的正则化器类别。