  Empirical likelihood is a very important nonparametric approach which is of
wide application. However, it is hard and even infeasible to calculate the
empirical log-likelihood ratio statistic with massive data. The main challenge
is the calculation of the Lagrange multiplier. This motivates us to develop a
distributed empirical likelihood method by calculating the Lagrange multiplier
in a multi-round distributed manner. It is shown that the distributed empirical
log-likelihood ratio statistic is asymptotically standard chi-squared under
some mild conditions. The proposed algorithm is communication-efficient and
achieves the desired accuracy in a few rounds. Further, the distributed
empirical likelihood method is extended to the case of Byzantine failures. A
machine selection algorithm is developed to identify the worker machines
without Byzantine failures such that the distributed empirical likelihood
method can be applied. The proposed methods are evaluated by numerical
simulations and illustrated with an analysis of airline on-time performance
study and a surface climate analysis of Yangtze River Economic Belt.
