Valid Inference for $L_2$-Boosting

We review several recently proposed post-selection inference frameworks and assess their transferability to the component-wise functional gradient descent algorithm (CFGD) under normality assumption for model errors, also known as $L_2$-Boosting. The CFGD is one of the most versatile toolboxes to analyze data, as it scales well to high-dimensional data sets, allows for a very flexible definition of additive regression models and incorporates inbuilt variable selection. %After addressing several issues associated with Due to the iterative nature, which can repeatedly select the same component to update, an inference framework for component-wise boosting algorithms requires adaptations of existing approaches; we propose tests and confidence intervals for linear, grouped and penalized additive model components estimated using the $L_2$-boosting selection process. We apply our framework to the prostate cancer data set and investigate the properties of our concepts in simulation studies. %The most general and promising selective inference framework for $L_2$-Boosting as well as for more general gradient-descent boosting algorithms is an sampling approach which constitutes an adoption of the recently proposed method by Yang et al. (2016).