Applied Modeling Techniques and Data Analysis 1

11 Weak Signals in High-Dimensional Poisson Regression Models

We addressed parameter estimation in the context of high-dimensional sparse Poisson regression models, in which the number of predictors exceeds the sample size. Generally, predictor screening via penalized maximum likelihood methods is required to provide the sparsity structure in the parsimonious model, before applying post-screening parameter estimation via maximum likelihood based on such a model. The major problem is that the use of different screening methods produces different sparsity structures, usually of unknown correctness. This may produce either overfitted or underfitted models, making post-screening maximum likelihood estimation based on these models inefficient. We therefore proposed post-screening estimation based on linear shrinkage, pretest and Stein-type shrinkage strategies to address inefficient maximum likelihood estimation based on the parsimonious models from the screening stage of unknown of appropriateness. Through Monte Carlo simulations, with unknown correctness in the predictor screening stage, the proposed estimators were shown to be significantly more efficient than the classical maximum likelihood estimators.

11.1. Introduction

The Poisson regression model is widely applied for predicting count response in fields including medicine, biotechnology and quality control; see Myers et al. (2012) or Agresti (2015) for more details. Maximum likelihood (ML) is a well-known statistical method ...