Pretest and shrinkage estimators in generalized partially linear models with application to real data

Semiparametric models hold promise to address many challenges to statistical inference that arise from real-world applications, but their novelty and theoretical complexity create challenges for estimation. Taking advantage of the broad applicability of semiparametric models, we propose some novel and improved methods to estimate the regression coefficients of generalized partially linear models (GPLM). This model extends the generalized linear model by adding a nonparametric component. Like in parametric models, variable selection is important in the GPLM to single out the inactive covariates for the response. Instead of deleting inactive covariates, our approach uses them as auxiliary information in the estimation procedure. We then define two models, one that includes all the covariates and another that includes the active covariates only. We then combine these two model estimators optimally to form the pretest and shrinkage estimators. Asymptotic properties are studied to derive the asymptotic biases and risks of the proposed estimators. We show that if the shrinkage dimension exceeds two, the asymptotic risks of the shrinkage estimators are strictly less than those of the full model estimators. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed estimation methods. We then apply our proposed methods to two real data sets. Our simulation and real data results show that the proposed estimators perform with higher accuracy and lower variability in the estimation of regression parameters for GPLM compared with competing estimation methods.