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. 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:按大小排序的L1范助估计器(SLOPE)是回归中常用的正则化方法,它能导致估计系数的聚合。也就是说,该估计器可以具有相同大小的系数。在本文中,我们推导了普通最小二乘、Huber和分位损失函数的SLOPE的渐近分布,并用它来研究其聚类行为。这需要更强的收敛性,因为聚类属性不仅仅来自于经典的弱收敛。我们建立了针对回归器的渐近正交设计的错误发现率的渐近控制。我们还展示了如何将框架扩展到除SLOPE以外的更广泛的正则化器类别中。