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

arXiv: https://arxiv.org/abs/2301.00139 · HTML: https://mlsanity.com/paper/2301.00139/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 44 references that are in the map.
- First posted on arXiv 31 Dec 2022.

## Builds on (most cited first, 7 of 7)

- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models](https://mlsanity.com/paper/1412.8765/) (2014)
- [Support recovery without incoherence: A case for nonconvex regularization](https://mlsanity.com/paper/1412.5632/) (2014)
- [Covariate Selection in High-Dimensional Generalized Linear Models With Measurement Error](https://mlsanity.com/paper/1407.1070/) (2014)
- [An l1,l2,l∞-Regularization Approach to High-Dimensional Errors-in-variables Models](https://mlsanity.com/paper/1412.7216/) (2014)
- [MEBoost: Variable Selection in the Presence of Measurement Error](https://mlsanity.com/paper/1701.02349/) (2017)
- [Inference in High-Dimensional Linear Measurement Error Models](https://mlsanity.com/paper/2001.10142/) (2020)

## Abstract (arXiv)

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.