  As a nonparametric statistical inference approach, empirical likelihood has
been found very useful in numerous occasions. However, it encounters serious
computational challenges when applied directly to the modern massive dataset.
This article studies empirical likelihood inference over decentralized
distributed networks, where the data are locally collected and stored by
different nodes. To fully utilize the data, this article fuses Lagrange
multipliers calculated in different nodes by employing a penalization
technique. The proposed distributed empirical log-likelihood ratio statistic
with Lagrange multipliers solved by the penalized function is asymptotically
standard chi-squared under regular conditions even for a divergent machine
number. Nevertheless, the optimization problem with the fused penalty is still
hard to solve in the decentralized distributed network. To address the problem,
two alternating direction method of multipliers (ADMM) based algorithms are
proposed, which both have simple node-based implementation schemes.
Theoretically, this article establishes convergence properties for proposed
algorithms, and further proves the linear convergence of the second algorithm
in some specific network structures. The proposed methods are evaluated by
numerical simulations and illustrated with analyses of census income and Ford
gobike datasets.
