Global debiased DC estimations for biased estimators via pro forma regression

We establish a global unbiased divide-and-conquer estimation (gub-DC) in linear model and a global bias reduced DC estimation (gbr-DC) in nonlinear model under the case of memory constraint. To introduce the new strategy in linear model, we first provide a new insight into the statistical structure through the closed representation of the local biased estimator and then construct a pro forma linear regression with the local estimator as “response variable” and the parameter of interest as “intercept.” Based on such a regression structure, we composite a global unbiased estimator as the least squares estimator of the intercept. Generally, the gub-DC method can be applied to various biased estimations such as Ridge estimator, principal component estimator and Stein estimator in linear model. Moreover, the method can be extended into nonlinear model to construct a global bias reduced estimator. The main advantage over the classical DC methods is that the new proposed procedures can absorb the information hidden in the statistical structure, and the resulting global estimators are strictly unbiased or can achieve root-n consistency, without any constraint on the number of batches. Another attractive feature refers to the computational simplicity and efficiency. Detailed simulation studies demonstrate that the new estimators are significantly bias-corrected, and their behaviors are comparable with the entire data estimation and are better or at least not worse than the competitors.