On high-dimensional Poisson models with measurement error: Hypothesis testing for nonlinear nonconvex optimization
Fei Jiang, Yeqing Zhou, Jianxuan Liu, Yanyuan Ma
The Annals of Statistics · 2023 · Volume 51 · Issue 1
Abstract
We study estimation and testing in the Poisson regression model with noisy high-dimensional covariates, which has wide applications in analyzing noisy big data. Correcting for the estimation bias due to the covariate noise leads to a nonconvex target function to minimize. Treating the high-dimensional issue further leads us to augment an amenable penalty term to the target function. We propose to estimate the regression parameter through minimizing the penalized target function. We derive the L1 and L2 convergence rates of the estimator and prove the variable selection consistency. We further establish the asymptotic normality of any subset of the parameters, where the subset can have infinitely many components as long as its cardinality grows sufficiently slow. We develop Wald and score tests based on the asymptotic normality of the estimator, which permits testing of linear functions of the members if the subset. We examine the finite sample performance of the proposed tests by extensive simulation. Finally, the proposed method is successfully applied to the Alzheimer’s Disease Neuroimaging Initiative study, which motivated this work initially.
Keywords
62F03, High-dimension inference, measurement error, nonconvex optimization, Poisson model
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