Sufficient Dimension Reduction for Poisson Regression

Abstract

Poisson regression is popular and commonly employed to analyze frequency of occurrences in a fixed amount of time. In practice, data collected from many scientific disciplines tend to grow in both volume and complexity. One characteristic of such complexity is the inherent sparsity in high-dimensional covariates space. Sufficient dimension reduction (SDR) is known to be an effective cure for its advantage of making use of all available covariates. Existing SDR techniques for a continuous or binary response do not naturally extend to count response data. It is challenging to detect the dependency between the response variable and the covariates due to the curse of dimensionality. To bridge the gap between SDR and its applications in count response models, an efficient estimating procedure is developed to recover the central subspace through estimating a finite dimensional parameter in a semiparametric model. The proposed model is flexible which does not require model assumption on the conditional mean or multivariate normality assumption on covariates. The resulting estimators achieve optimal semiparametric efficiency without imposing linearity or constant variance assumptions. The finite sample performance of the estimators is examined via simulations, and the proposed method is further demonstrated in the baseball hitter example and pathways to desistance study.