On High dimensional Poisson models with measurement error: hypothesis testing for nonlinear nonconvex optimization

TL;DR

This paper develops methods for estimation and hypothesis testing in high-dimensional Poisson regression models with noisy covariates, addressing bias correction, variable selection, and asymptotic inference.

Contribution

It introduces a penalized estimation approach for high-dimensional Poisson models with measurement error, establishing convergence rates, variable selection consistency, and asymptotic normality.

Findings

Proposed estimators achieve optimal convergence rates.

Variable selection consistency is established.

Finite sample tests perform well in simulations and real data.

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 non-convex 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.