Title: Penalized robust estimators in logistic regression with applications to sparse models Show Chinese title

Title: 带惩罚的鲁棒估计量在逻辑回归中的应用：稀疏模型

Abstract: Sparse covariates are frequent in classification and regression problems and in these settings the task of variable selection is usually of interest. As it is well known, sparse statistical models correspond to situations where there are only a small number of non--zero parameters and for that reason, they are much easier to interpret than dense ones. In this paper, we focus on the logistic regression model and our aim is to address robust and penalized estimation for the regression parameter. We introduce a family of penalized weighted $M-$type estimators for the logistic regression parameter that are stable against atypical data. We explore different penalizations functions and we introduce the so--called Sign penalization. This new penalty has the advantage that it depends only on one penalty parameter, avoiding arbitrary tuning constants. We discuss the variable selection capability of the given proposals as well as their asymptotic behaviour. Through a numerical study, we compare the finite sample performance of the proposal corresponding to different penalized estimators either robust or classical, under different scenarios. A robust cross--validation criterion is also presented. The analysis of two real data sets enables to investigate the stability of the penalized estimators to the presence of outliers. Show Chinese abstract

Abstract: 稀疏协变量在分类和回归问题中很常见，在这些情况下，变量选择通常是一个重要的任务。 众所周知，稀疏统计模型对应于只有少量非零参数的情况，因此它们比密集模型更容易解释。 本文聚焦于逻辑回归模型，并旨在解决回归参数的稳健性和惩罚性估计问题。 我们引入了一类针对逻辑回归参数的惩罚加权$M-$型估计量，这些估计量能够抵抗异常数据的影响。 我们探讨了不同的惩罚函数，并提出了所谓的“Sign惩罚”。 这种新惩罚的优势在于它仅依赖一个惩罚参数，避免了任意调整常数。 我们讨论了所提出方法的变量选择能力及其渐近行为。 通过数值研究，我们在不同场景下比较了不同惩罚估计量（无论是稳健的还是经典的）在有限样本下的性能。 此外，还提出了一个稳健的交叉验证准则。 通过对两个真实数据集的分析，可以研究惩罚估计量对异常值存在的稳定性。