  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.
