Variable screening in ultra-high dimensional linear regressions

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2022-05
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Wang, Run
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Dutta, Somak
Roy, Vivekananda
Dai, Xiongtao
Nordman, Daniel
Wang, Lily
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Statistics
Abstract
This dissertation is composed of three research projects focused on variable screening methods in ultra-high dimensional linear regressions. The first project points out several potential issues with the marginal (Pearson) correlation based screening method in the presence of correlated predictors. We use simulation studies and a real data example to demonstrate that when important variables are correlated, the marginal correlation based screening method may not be screening consistent. This study can serve as a warning against using such marginal correlation screening methods without further investigation. In the second project, we propose a novel Bayesian iterative screening (BITS) in ultra-high dimensional linear regressions. The BITS method can incorporate prior information on effect sizes and the number of true variables while not requiring more computational cost than many frequentist screening methods. Under mild conditions, we prove that BITS is screening consistent even under a more general q-exponential tail condition on the errors. We also present a new stopping rule based on the posterior probability which is screening consistent under the q-exponential tail condition on the errors. In the third project, we develop a new variable screening method in ultra-high dimensional linear regressions based on the ridge partial correlation (RPC) between the response and the predictors. The RPC is a ridge penalized partial correlation and can be computed by a fast Cholesky parallel algorithm. The computational cost of the RPC screening is no more than several other frequentist screening methods. Like BITS, we prove that the RPC screening is screening consistent under the q-exponential tail condition for errors. Simulation studies and a real data example show the fine screening performance of the RPC screening method.
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