Advisor: Mathias Drton & Ali Shojaie

Modern statistical problems are increasingly high-dimensional, with the number of covariables p potentially vastly exceeding sample size N. Fortunately, significant progress has been made in developing rigorous statistical tools for tackling such problems, but these methods have primarily targeted prediction, point estimation, and or variable selection.

There have been some proposals that address our need to also be able to assign uncertainty, statistical significance and confidence in high-dimensional models, which is of extreme practical importance when interpretation of parameters and variables is key. In this talk, we expand upon existing work by proposing an inferential framework for linear mixed effect models, which has not been previously addressed. Our goal is to be able to test null hypotheses of the form H0,j: b_j = 0 and H0,G: b_G = 0, b is the vector of fixed effect coefficients. Note that G can be any subset of {1, ... p}.

The framework we propose is inspired by the de-biased ridge estimator proposed in Buhlmann (2013). We develop theory and show that the framework yields asymptotically valid tests and confidence intervals. We show via numerical experiments that the method sufficiently controls for Type I error in hypothesis testing and generates confidence intervals that achieve target coverage, outperforming competitors that assume observations are homogeneous when in fact, they are correlated within group. Lastly, we demonstrate practical applicability of our method using the riboflavin production dataset from Buhlmann et al. (2014).