  We propose a distributed method for simultaneous inference for datasets with
sample size much larger than the number of covariates, i.e., N >> p, in the
generalized linear models framework. When such datasets are too big to be
analyzed entirely by a single centralized computer, or when datasets are
already stored in distributed database systems, the strategy of
divide-and-combine has been the method of choice for scalability. Due to
partition, the sub-dataset sample sizes may be uneven and some possibly close
to p, which calls for regularization techniques to improve numerical stability.
However, there is a lack of clear theoretical justification and practical
guidelines to combine results obtained from separate regularized estimators,
especially when the final objective is simultaneous inference for a group of
regression parameters. In this paper, we develop a strategy to combine
bias-corrected lasso-type estimates by using confidence distributions. We show
that the resulting combined estimator achieves the same estimation efficiency
as that of the maximum likelihood estimator using the centralized data. As
demonstrated by simulated and real data examples, our divide-and-combine method
yields nearly identical inference as the centralized benchmark.
