Sparsified Simultaneous Confidence Intervals for High-Dimensional Linear Models

High-dimensional data analysis plays an important role in modern scientific discoveries. There has been extensive work on high-dimensional variable selection and estimation using penalized regressions, such as Lasso (Tibshirani, 1996), SCAD (Fan and Li, 2001), MCP (Zhang et al., 2010), and selection by partitioning solution paths (Liu and Wang, 2018). In recent years, inference for the true regression coefficients and the true model began to attract attention. A major challenge of high-dimensional inference is how to quantify the uncertainty of the coefficient estimate because such uncertainty depends on two components, the uncertainty in parameter estimation given the selected model, the uncertainty in selecting the model, both of which are difficult to estimate and are actively studied. For inference of the regression coefficients, Scheffé (1953) introduces the notion of simultaneous confidence intervals, which is a sequence of intervals containing the true coefficients at a given probability. For the high-dimensional linear models, Dezeure et al. (2017) and Zhang and Cheng (2017) construct the simultaneous confidence intervals using the debiased Lasso approach (van de Geer et al., 2014; Zhang and Zhang, 2014).