Poisson Regression With Error Corrupted High Dimensional Features

Abstract

Features extracted from aggregated data are often contaminated with errors. Errors in these features are usually difficult to handle, especially so when the feature dimension is high. We construct an estimator of the feature effects in the context of Poisson regression with high dimensional feature and additive measurement errors. The procedure is based on penalizing a target function that is specially designed to handle measurement errors. We perform optimization within a bounded region. Benefitting from the convexity of the constructed target function in this region, we establish the theoretical properties of the new estimator in terms of algorithmic convergence and statistical consistency. Numerical performance is demonstrated through simulation studies. We apply the method to analyze the possible effect of weather on the number of COVID-19 cases.

Overview of High-Dimensional Measurement Error Regression Models

“…In order to analyze count data, Poisson regression models are a popular choice in practice. Jiang and Ma [78] studied the high-dimensional Poisson regression models with additive measurement errors and proposed a novel optimization algorithm to obtain the estimator of true regression coefficient vector β 0 . Suppose that Y i is the response variable following a Poisson distribution satisfying E(Y i |X i ) = exp(X T i β), where X i ∈ R p is an unobservable covariate.…”

“…From ( 35), Jiang and Ma [78] imposed a restriction on β similar to it in [63] and estimated β by solving the following optimization problem…”

“…Bai et al [77] proposed a variable selection method for ultrahigh-dimensional linear quantile regression models with measurement errors. Jiang and Ma [78] drew on the idea of nonconvex Lasso in [63] and proposed an estimator of the regression coefficients for high-dimensional Poisson models with measurement errors. Byrd and McGee [79] developed an iterative estimation method for high-dimensional generalized linear models with additive measurement errors based on the imputationregularized optimization (IRO) algorithm in [80].…”

The naïve estimator of a Poisson regression model with measurement errors

“…Approaches to a Poisson regression model with classical errors have been discussed by Kukush et al [1], Shklyar and Schneeweiss [8], Jiang and Ma [9], Guo and Li [10], and so on. Kukush et al [1] described the statistical properties of the naïve estimator, corrected score estimator, and structural quasi score estimator of a Poisson regression model with normally distributed explanatory variable and measurement error.…”

“…Shklyar and Schneeweiss [8] assumed an explanatory variable and a measurement error with a multivariate normal distribution and compared the asymptotic covariance matrices of the corrected score estimator, simple structural estimator, and structural quasi score estimator of a Poisson regression model. Jiang and Ma [9] assumed a high-dimensional explanatory variable with a multivariate normal error and proposed a new estimator for a Poisson regression model by combining Lasso regression and the corrected score function. Guo and Li [10] assumed a Poisson regression model with classical errors and proposed an estimator that is a generalization of the corrected score function discussed in Nakamura [7] for generally distributed errors; they derived the asymptotic normality of the proposed estimator.…”