High-Dimensional Inference beyond Linear Models
This project concerns making proper statistical inference for high-dimensional parameters in three sets of widely used regression models: (i) generalized linear models; (ii) Cox models for censored univariate, multivariate, or clustered survival data; and (iii) functional regression with three-dimensional functional inputs. The current literature has primarily focused on linear regression models with high-dimensional covariates. One type of method is the so-called post-selection inference conditional on the selected model. Another type of method parallel to post-selection inference is to correct the biases of lasso estimates in the full model, the so-called de-biased lasso or de-biased lasso, which has been shown to possess nice theoretical and numerical properties in linear regression models. The assumptions for de-biased lasso in linear models have been directly applied to nonlinear models, e.g., generalized linear models and the Cox model for survival data, in the current literature. We find, however, that the key sparsity assumption for the inverse expected Hessian matrix hardly holds even when the precision matrix of the covariates is indeed sparse, an important sufficient condition for the de-biased approach to work in linear regression models. In this project, we will investigate new methods that we call refined de-biased methods for all three different sets of models mentioned above by further de-biasing without imposing the sparsity matrix assumption. Each set of models possesses its unique challenges.