Robust estimators for Generalized Linear Models (GLMs) are not easy to develop because of the nature of the distributions involved. Recently, there has been an increasing interest in this topic, especially in the presence of a possibly large number of explanatory variables. Transformed M-estimators (MT) are a natural way to extend the methodology of M-estimators to the class of GLMs and to obtain robust methods. We introduce a penalized version of MT-estimators in order to deal with high-dimensional data. We prove, under appropriate assumptions, consistency and asymptotic normality of this new class of estimators. The theory is developed for redescending $\rho$-functions and Elastic Net penalization. An iterative re-weighted least squares algorithm is given, together with a procedure to initialize it. The latter is of particular importance, since the estimating equations might have multiple roots. We illustrate the performance of this new method for the Poisson family under several type of contaminations in a Monte Carlo experiment and in an example based on a real dataset.

翻译：广义线性模型的稳健估计量因所涉分布性质而难以发展。近年来，这一课题日益受到关注，尤其是在存在大量解释变量的情况下。变换M估计量是将M估计量方法论扩展到广义线性模型类别并获得稳健方法的自然途径。我们引入了一种带罚项的变换M估计量，以处理高维数据。在适当假设下，我们证明了这类新估计量的一致性和渐近正态性。该理论针对下降型ρ函数和弹性网络惩罚进行展开。我们给出了迭代重加权最小二乘算法及其初始化程序，后者尤为重要，因为估计方程可能存在多个根。我们通过蒙特卡洛实验和基于真实数据集的示例，展示了该新方法在泊松族受多种污染情况下的性能。