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

Count data are routinely encountered in practice. For example, cognitive scores in a neuroscience study, the number of deaths in an infectious disease study, and the number of clicks on a particular product on an e-commerce platform, are all count data. Because most of the count data are concentrated on a few small discrete values rather than expanded on the entire real line and because the distribution of count variables is often skewed, the familiar linear model becomes less ideal to capture these features. In the literature, Poisson regression (McCullagh & Nelder 2019) is arguably the most popular model to describe count outcomes, because it naturally models the skewed distribution for positive outcomes. On the other hand, together with the count data, a large number of covariates are often collected thanks to the ever advancing capability of modern technologies. However, these covariates are often contaminated with errors due to imperfect data acquisition and processing procedures. Ignoring these errors can produce biased results, which can finally lead to misleading statistical inference on the model parameters (Carroll et al. 2006) that explain the association between covariates and outcomes. Our goal is to develop rigorous statistical inference procedures to test linear hypotheses in the high dimensional Poisson model with noisy covariates. Such inference tools will enable explaining the association between the count outcome and the individual covariate or combination of covariate, quantifying the un-MSC2020 subject classifications: Primary 00X00, 00X00; secondary 00X00.